INTEGRALS CONTAINING Sinh(ax)

Nivedh Iyer

friends i hav come across many JEE papers in which these types of integration sums r not been able to b solved by the students ................ plzz refer the following formulae.............it will surely b useful.........do rate me if u like it.........!!!!!!!!

plzz do rate only those wch u think was useful.............!!!!!!!!!!

Nivedh Iyer

INTEGRALS CONTAINING Sinh(ax)


1.
2.: $displaystyleintsinh axdx=displaystyle rac{acosh ax}{a}-displaystyle rac{sinh ax}{a^2}$
3.: $displaystyleint x^2sinh axdx=left(displaystyle rac{x^2}{a}+displaystyle rac{2}{a^3} ight)cosh ax-displaystyle rac{2x}{a^2}sinh ax$
4.: $displaystyleintdisplaystyle rac{sinh ax}{x}dx=ax+displaystyle rac{(ax)^3}{3cdot 3!}+displaystyle rac{(ax)^5}{5cdot 5!}+cdotcdotcdot$
5.: $displaystyleintdisplaystyle rac{sinh ax}{x^2}dx=-displaystyle rac{sinh ax}{x}+aintdisplaystyle rac{cosh ax}{x}dx$
6.
7.: $displaystyleintdisplaystyle rac{xdx}{sinh ax}=displaystyle rac{1}{a^2}... ...tyle rac{2(-1)^n(2^{2n}-1)B_{n}(ax)^{2n+1}}{(2n+1)!}+cdotcdotcdot ight}$
8.: $displaystyleintsinh^2 axdx=displaystyle rac{sinh axcosh ax}{2a}-displaystyle rac{x}{2}$
9.: $displaystyleint xsinh^2 axdx=displaystyle rac{xsinh 2ax}{4a}-displaystyle rac{cosh 2ax}{8a^2}-displaystyle rac{x^2}{4}$
10.: $displaystyleintdisplaystyle rac{dx}{sinh^2 ax}=-displaystyle rac{coth ax}{a}$
11.
12.: $displaystyleintsinh axsinh pxdx=displaystyle rac{acosh axsin px-psinh axcos px}{a^2 + p^2}$
13.: $displaystyleintsinh ax cos pxdx=displaystyle rac{acosh ax cos px +psinh axsin px}{a^2 + p^2}$

14.: $displaystyleintdisplaystyle rac{dx}{p+qsinh ax}=displaystyle rac{1}{a... ...+p-displaystyle sqrt{p^2+q^2}}{qe^{ax}+p+displaystyle sqrt{p^2+q^2}} ight)$
15.: $displaystyleintdisplaystyle rac{dx}{(p+qsinh ax)^2}=displaystyle rac{-... ...nh ax)}+displaystyle rac{p}{p^2+q^2}intdisplaystyle rac{dx}{p+qsinh ax}$
16.: $displaystyleintdisplaystyle rac{dx}{p^2+q^2sinh^2 ax}=left{ egin{arra... ...2} anh ax}{p-displaystyle sqrt{p^2-q^2} anh ax} ight) end{array} ight.$
17.: $displaystyleintdisplaystyle rac{dx}{p^2-q^2sinh^2 ax}=displaystyle rac... ...laystyle sqrt{p^2+q^2} anh ax}{p-displaystyle sqrt{p^2+q^2} anh ax} ight)$
18.: $displaystyleint x^msinh ax dx=displaystyle rac{x^mcosh ax}{a}-displaystyle rac{m}{a}int x^{m-1}cosh axdx$
19.: $displaystyleintsinh^n axdx=displaystyle rac{sinh^{n-1}axcosh ax}{an}-displaystyle rac{n-1}{n}intsinh^{n-2}axdx$
20.: $displaystyleintdisplaystyle rac{sinh ax}{x^n}dx=displaystyle rac{-sin... ...^{n-1}}+displaystyle rac{a}{n-1}intdisplaystyle rac{cosh ax}{x^{n-1}}dx$
21.: $displaystyleintdisplaystyle rac{dx}{sinh^n ax}=displaystyle rac{-cosh... ...n-1}ax}-displaystyle rac{n-2}{n-1}intdisplaystyle rac{dx}{sinh^{n-2}ax}$
22.: $displaystyleintdisplaystyle rac{xdx}{sinh^n ax}=displaystyle rac{-xco... ...-2}ax}-displaystyle rac{n-2}{n-1}intdisplaystyle rac{xdx}{sinh^{n-2}ax}$

Nivedh Iyer

INTEGRALS CONTAINING Cosh(ax)

1.
2.: $displaystyleint xcosh axdx=displaystyle rac{xsinh ax}{a}-displaystyle rac{cosh ax}{a^2}$
3.: $displaystyleint x^2cosh axdx=-displaystyle rac{2xcosh ax}{a^2}+left( displaystyle rac{x^2}{a}+displaystyle rac{2}{a^3} ight) sinh ax$
4.: $displaystyleintdisplaystyle rac{cosh ax}{x}dx=ln x+displaystyle rac{(... ... rac{(ax)^4}{4cdot 4!}+displaystyle rac{(ax)^6}{6cdot 6!}+cdotcdotcdot$
5.: $displaystyleintdisplaystyle rac{cosh ax}{x^2}dx=-displaystyle rac{cosh ax}{x}+aintdisplaystyle rac{sinh ax}{x}dx$
6.: $displaystyleintdisplaystyle rac{dx}{cosh ax}=displaystyle rac{2}{a} an^{-1e^{ax}}$
7.: $displaystyleintdisplaystyle rac{x dx}{cosh ax}=displaystyle rac{1}{a^2... ...playstyle rac{(-1)^n E_{n}(ax)^{2n+2}}{(2n+2)(2n)!}+ cdotcdotcdot ight}$
where the constants E_n are the EULERS NOS..........!!!!!!
8.: $displaystyleintcosh^2 ax dx=displaystyle rac{x}{2}+displaystyle rac{sinh ax cosh ax}{2}$
9.: $displaystyleint xcosh^2 ax dx=displaystyle rac{x^2}{4}+displaystyle rac{xsinh 2ax}{4a}-displaystyle rac{cosh 2ax}{8a^2}$
10.: $displaystyleintdisplaystyle rac{dx}{cosh^2 ax}=displaystyle rac{ anh ax}{a}$
11.
12.: $displaystyleintcosh axsin px dx=displaystyle rac{asinh axsin px -pcosh axcos px}{a^2+p^2}$
13.: $displaystyleintcosh ax cos pxdx=displaystyle rac{asinh axcos px+pcosh axsin px}{a^2+p^2}$
14.
15.
16.: $displaystyleintdisplaystyle rac{x dx}{cosh ax+1}=displaystyle rac{x}{a... ...tyle rac{ax}{2}-displaystyle rac{2}{a^2}lncoshdisplaystyle rac{ax}{2}$
17.: $displaystyleintdisplaystyle rac{x dx}{cosh ax-1}=-displaystyle rac{x}{... ...tyle rac{ax}{2}+displaystyle rac{2}{a^2}lnsinhdisplaystyle rac{ax}{2}$
TAKE A BREAK.......now continue................!!!!!!!!!

18.: $displaystyleintdisplaystyle rac{dx}{(cosh ax+1)^2}=displaystyle rac{1}... ...ystyle rac{ax}{2}-displaystyle rac{1}{6a} anh^3displaystyle rac{ax}{2}$
19.: $displaystyleintdisplaystyle rac{dx}{(cosh ax-1)^2}=displaystyle rac{1}... ...ystyle rac{ax}{2}-displaystyle rac{1}{6a}coth^3displaystyle rac{ax}{2}$
20.: $displaystyleintdisplaystyle rac{dx}{p+qcosh ax}=left{ egin{array}{ll}... ...rt{p^2-q^2}}{qe^{ax}+p+displaystyle sqrt{p^2-q^2}} ight) end{array} ight.$
21.: $displaystyleintdisplaystyle rac{dx}{(p+qcosh ax)^2}=displaystyle rac{q... ...sh ax)}-displaystyle rac{p}{q^2-p^2}intdisplaystyle rac{dx}{p+qcosh ax}$
22.: $displaystyleintdisplaystyle rac{dx}{p^2-q^2cosh^2 ax}=left{ egin{arra... ...isplaystyle rac{p anh ax}{displaystyle sqrt{q^2-p^2}} end{array} ight.$
23.: $displaystyleintdisplaystyle rac{dx}{p^2+q^2cosh^2 ax}=left{ egin{arra... ...displaystyle rac{p anh ax}{displaystyle sqrt{p^2+q^2}} end{array} ight.$
24.: $displaystyleint x^m cosh ax dx=displaystyle rac{x^m sinh ax}{a}-displaystyle rac{m}{a}int x^{m-1}sinh ax dx$
25.: $displaystyleintcosh^n ax dx=displaystyle rac{cosh^{n-1}axsinh ax}{an}+displaystyle rac{n-1}{n}intcosh^{n-2} ax dx$
26.: $displaystyleintdisplaystyle rac{cosh ax}{x^n}dx=displaystyle rac{-cos... ...^{n-1}}+displaystyle rac{a}{n-1}intdisplaystyle rac{sinh ax}{x^{n-1}}dx$
27.: $displaystyleintdisplaystyle rac{dx}{cosh^n ax}=displaystyle rac{sinh ... ...n-1}ax}+displaystyle rac{n-2}{n-1}intdisplaystyle rac{dx}{cosh^{n-2}ax}$
28.: $displaystyleintdisplaystyle rac{x dx}{cosh^n ax}=displaystyle rac{xsi... ...2}ax}+displaystyle rac{n-2}{n-1}intdisplaystyle rac{x dx}{cosh^{n-2}ax}$

Nivedh Iyer

INTEGRALS CONTAINING ln(ax)

1.: $\displaystyle\int \ln xdx=x\ln x-x$
2.: $\displaystyle\int x\ln x dx=\displaystyle \frac{x^2}{2}(\ln x-\displaystyle \frac{1}{2})$
3.: $\displaystyle\int x^m\ln xdx=\displaystyle \frac{x^{m+1}}{m+1}\left(\ln x-\displaystyle \frac{1}{m+1}\right)$
4.: $\displaystyle\int\displaystyle \frac{\ln x}{x}dx=\displaystyle \frac{1}{2}\ln^2 x$
5.: $\displaystyle\int\displaystyle \frac{\ln x}{x^2}dx=-\displaystyle \frac{\ln x}{x}-\displaystyle \frac{1}{x}$
6.: $\displaystyle\int\ln^2 xdx=x\ln^2 x-2x\ln x+2x$
7.: $\displaystyle\int\displaystyle \frac{\ln^n xdx}{x}=\displaystyle \frac{\ln^{n+1}x}{n+1}$
8.: $\displaystyle\int\displaystyle \frac{dx}{x\ln x}=\ln (\ln x)$
9.: $\displaystyle\int\displaystyle \frac{dx}{\ln x}=\ln (\ln x)+\ln x+\displaystyle \frac{\ln^2 x}{2\cdot 2!}+\displaystyle \frac{\ln^3 x}{3\cdot 3!}+\cdot\cdot\cdot$
10.: $\displaystyle\int\displaystyle \frac{x^m dx}{\ln x}=\ln (\ln x)+(m+1)\ln x + \d... ...n^2 x}{2\cdot 2!}+\displaystyle \frac{(m+1)^3\ln^3x}{3\cdot 3!}+\cdot\cdot\cdot$
11.: $\displaystyle\int\ln^n xdx=x\ln^n x-n\int\ln^{n-1}xdx$
12.: $\displaystyle\int x^m\ln^n xdx=\displaystyle \frac{x^{m+1}\ln^n x}{m+1}-\displaystyle \frac{n}{m+1}\int x^m\ln^{n-1}xdx$
13.: $\displaystyle\int\ln(x^2+a^2)dx=x\ln(x^2+a^2)-2x+2a\tan^{-1}\displaystyle \frac{x}{a}$
14.: $\displaystyle\int\ln(x^2-a^2)dx=x\ln(x^2-a^2)-2x+a\ln\left(\displaystyle \frac{x+a}{x-a}\right)$
15.: $\displaystyle\int x^m\ln(x^2\pm a^2)dx=\displaystyle \frac{x^{m+1}\ln(x^2\pm a^2)}{m+1}-\displaystyle \frac{2}{m+1}\int\displaystyle \frac{x^{m+2}}{x^2\pm a^2}dx$

Nivedh Iyer

INTEGRALS CONTAINING e^ax

1.: $\displaystyle\int e^{ax} dx =\displaystyle \frac{e^{ax}}{a}$
2.: $\displaystyle\int xe^{ax}dx=\displaystyle \frac{e^{ax}}{a}\left(x-\displaystyle \frac{1}{a}\right)$
3.: $\displaystyle\int x^2 e^{ax}dx=\displaystyle \frac{e^{ax}}{a}\left(x^2-\displaystyle \frac{2x}{a}+\displaystyle \frac{2}{a^2}\right)$
4.: $\begin{array}{lcl} \displaystyle\int x^n e^{ax}dx&=& \displaystyle \frac{x^n e^... ...2}}{a^2}-\cdot\cdot\cdot \displaystyle \frac{(-1)^n n!}{a^n}\right) \end{array}$
5.: $\displaystyle\int\displaystyle \frac{e^{ax}}{x}dx=\ln x+\displaystyle \frac{ax}... ...\frac{(ax)^2}{2\cdot 2!}+\displaystyle \frac{(ax)^3}{3\cdot 3!}+\cdot\cdot\cdot$
6.: $\displaystyle\int\displaystyle \frac{e^{ax}}{x^n}dx=\displaystyle \frac{-e^{ax}... ...)x^{n-1}}+\displaystyle \frac{a}{n-1}\int\displaystyle \frac{e^{ax}}{x^{n-1}}dx$
7.: $\displaystyle\int\displaystyle \frac{dx}{p+qe^{ax}}=\displaystyle \frac{x}{p}-\displaystyle \frac{1}{ap}\ln (p+qe^{ax})$
8.: $\displaystyle\int\displaystyle \frac{dx}{(p+qe^{ax})^2}=\displaystyle \frac{x}{... ...displaystyle \frac{1}{ap(p+qe^{ax})}-\displaystyle \frac{1}{ap^2}\ln(p+qe^{ax})$
9.: $\displaystyle\int\displaystyle \frac{dx}{pe^{ax}+qe^{-ax}}=\left\{ \begin{array... ...style \sqrt{-q/p}}{e^{ax}+\displaystyle \sqrt{-q/p}}\right) \end{array}\right.$
10.: $\displaystyle\int e^{ax}\sin bx dx=\displaystyle \frac{e^{ax}(a\sin bx -b\cos bx)}{a^2+b^2}$
11.: $\displaystyle\int e^{ax}\cos bx dx=e^{ax}\displaystyle \frac{(a\cos bx+b\sin bx)}{a^2+b^2}$
12.: $\displaystyle\int xe^{ax}\sin bx dx=\displaystyle \frac{xe^{ax}(a\sin bx -b\cos... ...splaystyle \frac{e^{ax}\left\{(a^2-b^2)\sin bx-2ab\cos bx\right\}}{(a^2+b^2)^2}$
13.: $\displaystyle\int xe^{ax}\cos bx dx=\displaystyle \frac{xe^{ax}(a\cos bx +b\sin... ...splaystyle \frac{e^{ax}\left\{(a^2-b^2)\cos bx+2ab\sin bx\right\}}{(a^2+b^2)^2}$
14.: $\displaystyle\int e^{ax}\ln xdx=\displaystyle \frac{e^{ax}\ln x}{a}-\displaystyle \frac{1}{a}\int\displaystyle \frac{e^{ax}}{x}dx$
15.: $\displaystyle\int e^{ax}\sin^n bxdx=\displaystyle \frac{e^{ax}\sin^{n-1}bx}{a^2... ...\cos bx) + \displaystyle \frac{n(n-1)b^2}{a^2+n^2b^2}\int e^{ax}\sin^{n-2}bx dx$
16.: $\displaystyle\int e^{ax}\cos^n bxdx=\displaystyle \frac{e^{ax}\cos^{n-1}bx}{a^2... ...\sin bx) + \displaystyle \frac{n(n-1)b^2}{a^2+n^2b^2}\int e^{ax}\cos^{n-2}bx dx$

rini s

thankyou!!!

Nivedh Iyer

Integrals with Inverse Trigonometric Functions

1.: $\displaystyle\int\sin^{-1}\displaystyle \frac{x}{a}dx=x\sin^{-1} \displaystyle \frac{x}{a}+\displaystyle \sqrt{a^2-x^2}$
2.: $\displaystyle\int x\sin^{-1}\displaystyle \frac{x}{a}dx=\left(\displaystyle \fr... ...\displaystyle \frac{x}{a}+\displaystyle \frac{x\displaystyle \sqrt{a^2-x^2}}{4}$
3.: $\displaystyle\int x^2\sin^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...yle \frac{x}{a}+\displaystyle \frac{(x^2+2a^2)\displaystyle \sqrt{a^2-x^2)}}{9}$
4.: $\displaystyle\int\displaystyle \frac{\sin^{-1}(x/a)}{x}dx=\displaystyle \frac{x... ...rac{1\cdot 3\cdot 5(x/a)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7} + \cdot \cdot \cdot$
5.: $\displaystyle\int\displaystyle \frac{\sin^{-1}(x/a)}{x^2}dx=-\displaystyle \fra... ...rac{1}{a}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{a^2-x^2}}{x}\right)$
6.: $\displaystyle\int\left(\sin^{-1}\displaystyle \frac{x}{a}\right)^2 dx=x\left(\s... ...a}\right)^2 -2x+2\displaystyle \sqrt{a^2-x^2}\sin^{-1}\displaystyle \frac{x}{a}$
7.: $\displaystyle\int\cos^{-1}\displaystyle \frac{x}{a}dx=x\cos^{-1}\displaystyle \frac{x}{a}-\displaystyle \sqrt{a^2-x^2}$
8.: $\displaystyle\int x\cos^{-1}\displaystyle \frac{x}{a}dx=\left(\displaystyle \fr... ...\displaystyle \frac{x}{a}-\displaystyle \frac{x\displaystyle \sqrt{a^2-x^2}}{4}$

9.: $\displaystyle\int x^2\cos^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...tyle \frac{x}{a}-\displaystyle \frac{(x^2+2a^2)\displaystyle \sqrt{a^2-x^2}}{9}$
10.: $\displaystyle\int\displaystyle \frac{\cos^{-1}(x/a)}{x}dx=\displaystyle \frac{\pi}{2}\ln x-\int\displaystyle \frac{\sin^{-1}(x/a)}{x}dx$
11.: $\displaystyle\int\displaystyle \frac{\cos^{-1}(x/a)}{x^2}dx=-\displaystyle \fra... ...rac{1}{a}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{a^2-x^2}}{x}\right)$
12.: $\displaystyle\int\left(\cos^{-1}\displaystyle \frac{x}{a}\right)^2 dx=x\left(\c... ...{a}\right)^2-2x-2\displaystyle \sqrt{a^2-x^2}\cos^{-1}\displaystyle \frac{x}{a}$
13.: $\displaystyle\int\tan^{-1}\displaystyle \frac{x}{a}dx=x\tan^{-1}\displaystyle \frac{x}{a}-\displaystyle \frac{a}{2}\ln(x^2+a^2)$
14.: $\displaystyle\int x\tan^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{1}{2}(x^2+a^2)\tan^{-1}\displaystyle \frac{x}{a}-\displaystyle \frac{ax}{2}$
15.: $\displaystyle\int x^2\tan^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...frac{x}{a}-\displaystyle \frac{ax^2}{6}+\displaystyle \frac{a^3}{6}\ln(x^2+a^2)$

16.: $\displaystyle\int\displaystyle \frac{\tan^{-1}(x/a)}{x}dx=\displaystyle \frac{x... ...playstyle \frac{(x/a)^5}{5^2}-\displaystyle \frac{(x/a)^7}{7^2}+\cdot\cdot\cdot$
17.: $\displaystyle\int\displaystyle \frac{\tan^{-1}(x/a)}{x^2}dx=-\displaystyle \fra... ...{a}-\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{x^2+a^2}{x^2}\right)$
18.: $\displaystyle\int\cot^{-1}\displaystyle \frac{x}{a}dx=x\cot^{-1}\displaystyle \frac{x}{a}+\displaystyle \frac{a}{2}\ln(x^2+a^2)$
19.: $\displaystyle\int x\cot^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{1}{2}(x^2+a^2)\cot^{-1}\displaystyle \frac{x}{a}+\displaystyle \frac{ax}{2}$
20.: $\displaystyle\int x^2\cot^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...frac{x}{a}+\displaystyle \frac{ax^2}{6}-\displaystyle \frac{a^3}{6}\ln(x^2+a^2)$
21.: $\displaystyle\int\displaystyle \frac{\cot^{-1}(x/a)}{x}dx=\displaystyle \frac{\pi}{2}\ln x-\int\displaystyle \frac{\tan^{-1}(x/a)}{x}dx$

22.: $\displaystyle\int\displaystyle \frac{\cot^{-1}(x/a)}{x^2}dx=-\displaystyle \fra... ...{x}+\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{x^2+a^2}{x^2}\right)$
23.: $\displaystyle\int\sec^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}{ll... ...style \frac{\pi}{2}<\sec^{-1}\displaystyle \frac{x}{a}<\pi \end{array}\right.$
24.: $\displaystyle\int x\sec^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}{l... ...ystyle \frac{\pi}{2}<\sec^{-1}\displaystyle \frac{x}{a}<\pi \end{array}\right.$
25.: $\displaystyle\int x^2\sec^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}... ...ystyle \frac{\pi}{2}<\sec^{-1}\displaystyle \frac{x}{a}<\pi \end{array}\right.$
26.: $\displaystyle\int\displaystyle \frac{\sec^{-1}(x/a)}{x}dx=\displaystyle \frac{\... ...\frac{1\cdot 3\cdot 5(a/x)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7} + \cdot\cdot\cdot$
27.: $\displaystyle\int\displaystyle \frac{\sec^{-1}(x/a)}{x^2}dx=\left\{ \begin{arra... ...ystyle \frac{\pi}{2}<\sec^{-1}\displaystyle \frac{x}{a}<\pi \end{array}\right.$
28.: $\displaystyle\int\csc^{-1}\displaystyle \frac{x}{a}dx=\left\{ \displaystyle\beg... ...laystyle \frac{\pi}{2}<\csc^{-1}\displaystyle \frac{x}{a}<0 \end{array}\right.$
29.: $\displaystyle\int x\csc^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}{l... ...laystyle \frac{\pi}{2}<\csc^{-1}\displaystyle \frac{x}{a}<0 \end{array}\right.$

30.: $\displaystyle\int x^2\csc^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}... ...laystyle \frac{\pi}{2}<\csc^{-1}\displaystyle \frac{x}{a}<0 \end{array}\right.$
31.: $\displaystyle\int\displaystyle \frac{\csc^{-1}(x/a)}{x}dx=-\left(\displaystyle ... ...{1\cdot 3\cdot 5(a/x)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7}+\cdot\cdot\cdot \right)$
32.: $\displaystyle\int\displaystyle \frac{\csc^{-1}(x/a)}{x^2}dx=\left\{ \begin{arra... ...laystyle \frac{\pi}{2}<\csc^{-1}\displaystyle \frac{x}{a}<0 \end{array}\right.$
33.: $\displaystyle\int x^m\sin^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...e \frac{1}{m+1}\int\displaystyle \frac{x^{m+1}}{\displaystyle \sqrt{a^2-x^2}}dx$
34.: $\displaystyle\int x^m\cos^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...e \frac{1}{m+1}\int\displaystyle \frac{x^{m+1}}{\displaystyle \sqrt{a^2-x^2}}dx$
35.: $\displaystyle\int x^m\tan^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...ac{x}{a}-\displaystyle \frac{a}{m+1}\int\displaystyle \frac{x^{m+1}}{x^2+a^2}dx$
36.: $\displaystyle\int x^m\cot^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...ac{x}{a}+\displaystyle \frac{a}{m+1}\int\displaystyle \frac{x^{m+1}}{x^2+a^2}dx$
37.: $\displaystyle\int x^m\sec^{-1}\displaystyle \frac{x}{a}=\left \{ \begin{array}{... ...aystyle \frac{\pi}{2}<sec^{-1}\displaystyle \frac{x}{a}<\pi \end{array}\right.$
38.: $\displaystyle\int x^m\csc^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}... ...laystyle \frac{\pi}{2}<\csc^{-1}\displaystyle \frac{x}{a}<0 \end{array}\right.$

Shobhit Nandkeolyar

great work nivedh_89.
I have rated for your work.
but it would have been better if you had posted it in the community shelf.

Nivedh Iyer

thanx and my name is nivedh...............not nivedh_89!!!!!!!!!!

diksha jeena

gr8 work nivedh

keep postin thing like this!!!!!!!!

Ankur Limaye

Great work yaar ........

Tu to mera calculus sudhravaega ........

Nivedh Iyer

DEFINITE INTEGRALS CONTAINING EXPONENTIAL FUNCTIONS

1.: $displaystyleint_{0}^{infty}e^{-ax}cos bx dx=displaystyle rac{a}{a^2+b^2}$
2.: $displaystyleint_{0}^{infty}e^{-ax}sin bx dx=displaystyle rac{b}{a^2+b^2}$
3.: $displaystyleint_{0}^{infty}displaystyle rac{e^{-ax}sin bx}{x}dx= an^{-1}displaystyle rac{b}{a}$
4.: $displaystyleint_{0}^{infty}displaystyle rac{e^{-ax}-e^{-bx}}{x}dx=lndisplaystyle rac{b}{a}$
5.: $displaystyleint_{0}^{infty}e^{-ax^2}dx=displaystyle rac{1}{2}displaystyle sqrt{displaystyle rac{pi}{a}}$
6.: $displaystyleint_{0}^{infty}e^{-ax^2}cos bx dx=displaystyle rac{1}{2}displaystyle sqrt{displaystyle rac{pi}{a}}e^{-b^2/4a}$
7.: $displaystyleint_{0}^{infty}e^{-(ax^2+bx+c)}dx=displaystyle rac{1}{2}displaystyle sqrt{displaystyle rac{pi}{a}}e^{(b^2-4ac)/4a}$
8.: $displaystyleint_{-infty}^{infty}e^{-(ax^2+bx+c)}dx=displaystyle sqrt{displaystyle rac{pi}{a}}e^{(b^2-4ac)/4a}$
9.: $displaystyleint_{0}^{infty}x^n e^{-ax}dx=displaystyle rac{Gamma(n+1)}{a^n+1}$
10.: $displaystyleint_{0}^{infty}x^m e^{-ax^2}dx=displaystyle rac{Gamma[(m+1)/2]}{2a^{(m+1)/2}}$
11.: $displaystyleint_{0}^{infty}e^{-(ax^2+b/x^2)}dx=displaystyle rac{1}{2}displaystyle sqrt{displaystyle rac{pi}{a}}e^{-2displaystyle sqrt{ab}}$
12.: $displaystyleint_{0}^{infty}displaystyle rac{xdx}{e^x-1}=displaystyle r... ...3^2}+displaystyle rac{1}{4^2}+cdotcdotcdot =displaystyle rac{pi^2}{6}$
13.: $displaystyleint_{0}^{infty}displaystyle rac{x^{n-1}}{e^x-1}dx=Gamma(n+1)... ...displaystyle rac{1}{2^n}+displaystyle rac{1}{3^n}+cdotcdotcdot ight)$
14.: $displaystyleint_{0}^{infty}displaystyle rac{xdx}{e^x+1}=displaystyle r... ...2}-displaystyle rac{1}{4^2}+ cdotcdotcdot =displaystyle rac{pi^2}{12}$
15.: $displaystyleint_{0}^{infty}displaystyle rac{x^{n-1}}{e^x+1}dx=Gamma(n+1)... ...displaystyle rac{1}{2^n}+displaystyle rac{1}{3^n}-cdotcdotcdot ight)$
16.: $displaystyleint_{0}^{infty}displaystyle rac{sin mx}{e^{2pi x}-1}dx=displaystyle rac{1}{4}cothdisplaystyle rac{m}{2}-displaystyle rac{1}{2m}$
17.: $displaystyleint_{0}^{infty}left( displaystyle rac{1}{1+x}-e^{-x} ight)displaystyle rac{dx}{x}=gamma$
where the constant is the eulers constant.
18.: $displaystyleint_{0}^{infty}displaystyle rac{e^{-x^2}-e^{-x}}{x}dx=displaystyle rac{1}{2}gamma$
where the constant is the EULERs CONSTANT.
19.: $displaystyleint_{0}^{infty}left( displaystyle rac{1}{e^x-1}-displaystyle rac{e^{-x}}{x} ight)dx=gamma$
where the constant is the EULERs CONSTANT.
20.: $displaystyleint_{0}^{infty}displaystyle rac{e^{-ax}-e^{-bx}}{xsec px}dx=displaystyle rac{1}{2}lnleft(displaystyle rac{b^2+p^2}{a^2+p^2} ight)$
21.: $displaystyleint_{0}^{infty}displaystyle rac{e^{-ax}-e^{-bx}}{xcsc px}dx= an^{-1}displaystyle rac{b}{p}- an^{-1}displaystyle rac{a}{p}$
22.: $displaystyleint_{0}^{infty}displaystyle rac{e^{-ax}(1-cos x)}{x^2}dx=cot^{-1}a-displaystyle rac{a}{2}ln(a^2+1)$

Nivedh Iyer

NOW..............

COMMON SUBSTITUTIONS

1.: $\displaystyle \int F(ax+b)dx = \displaystyle \frac{1}{a} \displaystyle \int F(u)du$
where $u=ax\,+\,b$
2.: $\displaystyle \int F\left(\displaystyle\sqrt{ax\, +\, b}\right)\, dx = \displaystyle \frac{2}{a} \displaystyle \int u\,F(u)\,du$
where $u=\displaystyle\sqrt{ax\,+\,b}$
3.: $\displaystyle \int F\left( \sqrt[n]{ax+b} \right) \,dx = \displaystyle \frac{n}{a} \displaystyle \int u^{n-1}\,F(u)\,du$
where $u=\sqrt[n]{ax+b}$
4.: $\displaystyle \int F\left( \displaystyle\sqrt{a^{2}-x^{2}}\right)\,dx = a\,\displaystyle \int F(a \cos u)\,\cos u\,du$
where $x=a\sin u$
5.: $\displaystyle \int F\left( \displaystyle\sqrt{x^2+a^{2}} \right)\,dx= a\,\displaystyle \int F(a \sec u) \sec ^{2} u \, du$
where $x=a\tan u$
6.: $\displaystyle \int F\left( \displaystyle\sqrt{x^{2}-a^{2}} \right)\,dx=a \displaystyle \int F(a\tan u) \sec u \tan u\,du$
where $x=a\sec u$
7.: $\displaystyle \int F (e\displaystyle^{ax})\,dx = \displaystyle \frac{1}{a} \displaystyle \int\displaystyle \frac{F(u)}{u}\,du$
where $u=e\displaystyle^{ax}$
8.: $\displaystyle \int F(\ln x)\,dx = \displaystyle \int F(u)\,e\displaystyle^u\,du$
where $u=\ln x$
9.: $\displaystyle \int F\left( \sin ^{-1}\displaystyle \frac{x}{a}\right)\,dx = a\,\displaystyle \int F(u)\cos u\,du$
where $u=\sin ^{-1}\displaystyle \frac{x}{a}$

Nivedh Iyer

wait some more coming............!!!!!!!!!!!!

Nivedh Iyer

Common Integrals

1.: $\displaystyle \int adx=ax$
2.: $\displaystyle \int af(x)dx=a \displaystyle \int f(x)dx$
3.: $\displaystyle \int \left( u \pm v \pm w \pm \cdots \right) dx = \displaystyle \int udx \pm \displaystyle \int vdx \pm \displaystyle \int wdx \pm \cdots$
4.: $\displaystyle \int udv = uv - \displaystyle \int vdu$
5.: $\displaystyle \int f(ax)dx = \displaystyle \frac{1}{a} \displaystyle \int f(u)du$
6.: $\displaystyle \int F\{f(x)\}dx = \displaystyle \int F(u) \displaystyle \frac{dx}{du}du = \displaystyle \int \displaystyle \frac{F(u)}{f'(x)}du$
7.: $\displaystyle \int u^{n}du = \displaystyle \frac{u^{n+1}}{n+1}, n \neq -1$
8.: $\begin{array}{lcl} \displaystyle \int\displaystyle \frac{du}{u} & = & \ln u \mb... ...or} \ln (-u) \mbox{ if} u<0 \\ & = & \ln \left\vert u \right\vert \end{array}$
9.: $\displaystyle \int e^{u}du=e^{u}$
10.: $\displaystyle \int a^{u}du = \int e^{u \ln a}du = \displaystyle \frac{e^{u \ln a}}{\ln a} = \displaystyle \frac{a^{u}}{\ln a} , a >0, a \neq 1$
11.: $\displaystyle \int \sin u du = -\cos u$
12.: $\displaystyle \int \cos u du = \sin u$
13.: $\displaystyle \int \tan u du = \ln \sec u = -\ln \cos u$
14.: $\displaystyle \int \cot u du = \ln \sin u$
15.: $\displaystyle \int \sec u du = \ln (\sec u + \tan u) = \ln \tan \left( \displaystyle \frac{u}{2} + \displaystyle \frac{\pi}{4} \right)$
16.: $\displaystyle \int \csc u du = \ln (\csc u - \cot u) = \ln \tan \displaystyle \frac{u}{2}$

17.: $\displaystyle \int \sec ^{2} u du = \tan u$
18.: $\displaystyle \int \csc ^{2} u du = -\cot u$
19.: $\displaystyle \int \tan ^{2} u du = \tan u - u$
20.: $\displaystyle \int \cot ^{2} u du = -\cot u - u$
21.: $\displaystyle \int \sin ^{2} u du = \displaystyle \frac{u}{2} - \displaystyle \frac{\sin 2u}{4} = \displaystyle \frac{1}{2} (u-\sin u \cos u)$
22.: $\displaystyle \int \cos ^{2} u du = \displaystyle \frac{u}{2} + \displaystyle \frac{\sin 2u}{4} = \displaystyle \frac{1}{2} (u+\sin u \cos u)$
23.: $\displaystyle \int \sec u \tan u du = \sec u$
24.: $\displaystyle \int \csc u \cot u du = -\csc u$
25.: $\displaystyle \int \sinh u du = \cosh u$
26.: $\displaystyle \int \cosh u du = \sinh u$

27.: $\displaystyle \int \tanh u du = \ln \cosh u$
28.: $\displaystyle \int \coth u du = \ln \sinh u$
29.: $\displaystyle \int$ sech $u du = \sin ^{-1}(\tanh u )$ or $2\tan ^{-1}e^{u}$
30.: $\displaystyle \int$ csch $u du = \ln \tanh \displaystyle \frac{u}{2}$ or $-\coth ^{-1}e^{u}$
31.: $\displaystyle \int$ sech $^{2} u du = \tanh u$
32.: $\displaystyle \int$ csch ² u du =-coth u
33.: $\displaystyle \int\tanh ^{2} u du = u - \tanh u$
34.: $\displaystyle \int$ coth ² u du = u -coth u
35.: $\displaystyle \int\sinh ^{2} u du = \displaystyle \frac{\sinh 2u}{4} - \displaystyle \frac{u}{2} = \displaystyle \frac{1}{2}(\sinh u \cosh u- u)$
36.: $\displaystyle \int\cosh ^{2} u du = \displaystyle \frac{\sinh 2u}{4} + \displaystyle \frac{u}{2} = \displaystyle \frac{1}{2}(\sinh u \cosh u+ u)$

37.: $\displaystyle \int$ sech $u \tanh u du = -$ sech u
38.: $\displaystyle \int$ csch ucoth u du = -csch u
39.: $\displaystyle \int\displaystyle \frac{du}{u^{2}+a^{2}} = \displaystyle \frac{1}{a} \tan^{-1} \displaystyle \frac{u}{a}$
40.: $\displaystyle \int\displaystyle \frac{du}{u^{2} - a^{2}}= \displaystyle \frac{1... ...n \left( \displaystyle \frac{u - a}{u+a} \right) = - \displaystyle \frac{1}{a}$ coth $^{-1} \displaystyle \frac{u}{a} , u^{2}>a^{2}$
41.: $\displaystyle \int\displaystyle \frac{du}{a^{2}-u^{2}}= \displaystyle \frac{1}{... ...= \displaystyle \frac{1}{a} \tanh ^{-1} \displaystyle \frac{u}{a} , u^{2}<a^{2}$
42.: $\displaystyle \int\displaystyle \frac{du}{\sqrt{a^{2}-u^{2}}} = \sin ^{-1} \displaystyle \frac{u}{a}$
43.: $\displaystyle \int\displaystyle \frac{du}{\sqrt{u^{2}+a^{2}}} = \ln \left( u+ \displaystyle\sqrt{u^{2} + a^{2}} \right)$ or $\sinh ^{-1} \displaystyle \frac{u}{a}$
44.: $\displaystyle \int\displaystyle \frac{du}{\sqrt{u^{2}-a^{2}}} = \ln \left( u + \displaystyle\sqrt{u^{2} - a^{2}} \right)$
45.: $\displaystyle \int\displaystyle \frac{du}{u \sqrt{u^{2}-a^{2}}} = \displaystyle \frac{1}{a} \sec ^{-1} \left\vert \displaystyle \frac{u}{a} \right\vert$
46.: $\displaystyle \int\displaystyle \frac{du}{u \sqrt{u^{2}+a^{2}}}=-\displaystyle \frac{1}{a} \ln \left( \displaystyle \frac{a+\sqrt{u^{2}+a^{2}}}{u} \right)$
47.: $\displaystyle \int\displaystyle \frac{du}{u \sqrt{a^{2}-u^{2}}}=-\displaystyle \frac{1}{a} \ln \left( \displaystyle \frac{a+\sqrt{a^{2}-u^{2}}}{u} \right)$
48.: $\displaystyle \int f^{(n)}gdx = f^{(n-1)}g - f^{(n-2)}g' + f^{(n-3)} g'' - \cdots (-1)^{n} \displaystyle \int fg^{(n)}dx$

Swashata Ghosh

PHEW what a work. It will HELP ME A LOT. THANKS A LOT. I voted u

Nivedh Iyer

now for trigo........................

Table of Trigonometric Identities

Reciprocal identities

Pythagorean Identities

Quotient Identities

Co-Function Identities

Even-Odd Identities

Sum-Difference Formulas

Double Angle Formulas

Power-Reducing/Half Angle Formulas

Sum-to-Product Formulas

Product-to-Sum Formulas

Nivedh Iyer


`0`	`0`	`0`	`1`	`0`


				`1`


		`1`	`0`	`(undefined)`

Nivedh Iyer

now for DIFFERENTIAL CALCULUS................!!!!!!!!!!!!!!

Nivedh Iyer

First and Second Order Differential Equations

First Order Differential equations

A first order differential equation is of the form:

Linear Equations:

The general general solution is given by

where

is called the integrating factor.

Separable Equations:

(1): Solve the equation g(y) = 0 which gives the constant solutions.
(2): The non-constant solutions are given by

Bernoulli Equations:

(1): Consider the new function .
(2): The new equation satisfied by v is
(3): Solve the new linear equation to find v.
(4): Back to the old function y through the substitution .
(5): If n > 1, add the solution y=0 to the ones you got in (4).

Homogenous Equations:

is homogeneous if the function f(x,y) is homogeneous, that is

By substitution, we consider the new function

The new differential equation satisfied by z is

which is a separable equation. The solutions are the constant ones f(1,z) - z =0 and the non-constant ones given by

Do not forget to go back to the old function y = xz.

Exact Equations:

is exact if

The condition of exactness insures the existence of a function F(x,y) such that

All the solutions are given by the implicit equation

Second Order Differential equations

Homogeneous Linear Equations with constant coefficients:

Write down the characteristic equation

(1): If and are distinct real numbers (this happens if ), then the general solution is
(2): If (which happens if ), then the general solution is
(3): If and are complex numbers (which happens if ), then the general solution is

where

that is

Non Homogeneous Linear Equations:

The general solution is given by

where

is a particular solution and

is the general solution of the associated homogeneous equation

In order to find

two major techniques were developed.

: Method of undetermined coefficients or Guessing Method
This method works for the equation

where a, b, and c are constant and

where is a polynomial function with degree n. In this case, we have

where

The constants and have to be determined. The power s is equal to 0 if is not a root of the characteristic equation. If is a simple root, then s=1 and s=2 if it is a double root.
Remark. If the nonhomogeneous term g(x) satisfies the following

where are of the forms cited above, then we split the original equation into N equations

then find a particular solution . A particular solution to the original equation is given by
: Method of Variation of Parameters
This method works as long as we know two linearly independent solutions of the homogeneous equation

Note that this method works regardless if the coefficients are constant or not. a particular solution as

where and are functions. From this, the method got its name.
The functions and are solutions to the system:

which implies

Therefore, we have

Euler-Cauchy Equations:

where b and c are constant numbers. By substitution, set

then the new equation satisfied by y(t) is

which is a second order differential equation with constant coefficients.

(1): Write down the characteristic equation
(2): If the roots and are distinct real numbers, then the general solution is given by
(2): If the roots and are equal ( ), then the general solution is
(3): If the roots and are complex numbers, then the general solution is

where and .

Ask & Discuss Questions with Community & Experts

INTEGRALS CONTAINING Sinh(ax)

INTEGRALS CONTAINING Cosh(ax)

INTEGRALS CONTAINING ln(ax)

INTEGRALS CONTAINING eax

Integrals with Inverse Trigonometric Functions

DEFINITE INTEGRALS CONTAINING EXPONENTIAL FUNCTIONS

COMMON SUBSTITUTIONS

Common Integrals

Table of Trigonometric Identities

First and Second Order Differential Equations

First Order Differential equations

Linear Equations:

Separable Equations:

Bernoulli Equations:

Homogenous Equations:

Exact Equations:

Second Order Differential equations

Homogeneous Linear Equations with constant coefficients:

Non Homogeneous Linear Equations:

Method of undetermined coefficients or Guessing Method

Method of Variation of Parameters

Euler-Cauchy Equations:

INTEGRALS CONTAINING e^ax