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Nivedh Iyer

i hav not posted it bcas i want some rates........i hav already got 213 points for this article.....!!!!!!!!!This is for those who havent seen it yet.....!!!!!!!!!!!!

DEFINITE INTEGRALS THAT CONTAIN TRIGONOMETRIC FUNCTIONS

Note that all the constant are positive.

1.: $displaystyleint_{0}^{pi}sin mx sin nx dx=left{ egin{array}{ll} displa... ...in} &mbox{if $m$space and $n$space integers and};; m=n end{array} ight.$
2.: $displaystyleint_{0}^{pi} cos mx cos nx dx=left{ egin{array}{ll} displ... ...in} &mbox{if $m$space and $n$space integers and};; m=n end{array} ight.$
3.: $displaystyleint_{0}^{pi}sin mx cos nx dx=left{ egin{array}{ll} displa... ... $m$space and $n$space integers and $m+n$space even} end{array} ight.$
4.: $displaystyleint_{0}^{pi/2} sin^2 x dx=int_{0}^{pi/2}cos^2 dx=displaystyle rac{pi}{4}$
5.: $displaystyleint_{0}^{pi/2}sin^{2m} x dx=int_{0}^{pi/2}cos^{2m} x dx=dis... ...cdot 4cdot 6cdot cdotcdotcdot 2m}left(displaystyle rac{pi}{2} ight)$ ,
m=1,2,...
6.: $displaystyleint_{0}^{pi/2}sin^{2m+1}x dx=int_{0}^{pi/2}cos^{2m+1}x dx=d... ...c{2cdot 4cdot 6cdotcdotcdot 2m}{1cdot 3cdot 5cdot cdotcdotcdot 2m+1}$ ,
m=1,2,...
7.: $displaystyleint_{0}^{pi/2}sin^{2p-1}x cos^{2q-1}x dx=displaystyle rac{Gamma(p)Gamma(q)}{2Gamma(p+q)}$
8.: $displaystyleint_{0}^{infty}displaystyle rac{sin px}{x}dx=left{ egin{... ...n} p>0 0&hspace{.3in} p=0 -pi/2&hspace{.3in} p<0 end{array} ight.$
9.: $displaystyleint_{0}^{infty}displaystyle rac{sin pxcos qx}{x}dx=left{ ... ...0 pi/2&hspace{.3in} 0<p<q pi/4&hspace{.3in} p=q>0 end{array} ight.$
10.: $displaystyleint_{0}^{infty}displaystyle rac{sin px sin qx}{x^2}dx=left... ...hspace{.3in} 0<pleq q pi q/2&hspace{.3in} pgeq q>0 end{array} ight.$
11.: $displaystyleint_{0}^{infty}displaystyle rac{sin^2 px}{x^2}dx=displaystyle rac{pi p}{2}$
12.: $displaystyleint_{0}^{infty}displaystyle rac{1-cos px}{x^2}dx=displaystyle rac{pi p}{2}$
13.: $displaystyleint_{0}^{infty}displaystyle rac{cos px-cos qx}{x}dx=lndisplaystyle rac{q}{p}$
14.: $displaystyleint_{0}^{infty}displaystyle rac{cos px-cos qx}{x^2}dx=displaystyle rac{pi(q-p)}{2}$
15.: $displaystyleint_{0}^{infty}displaystyle rac{cos mx}{x^2+a^2}dx=displaystyle rac{pi}{2a}e^{-ma}$
16.: $displaystyleint_{0}^{infty}displaystyle rac{xsin mx}{x^2+a^2}dx=displaystyle rac{pi}{2}e^{-ma}$
17.: $displaystyleint_{0}^{infty}displaystyle rac{sin mx}{x(x^2+a^2)}dx=displaystyle rac{pi}{2a^2}(1-e^{-ma})$
18.: $displaystyleint_{0}^{2pi}displaystyle rac{dx}{a+bsin x}=displaystyle rac{2pi}{displaystyle sqrt{a^2-b^2}}$
19.: $displaystyleint_{0}^{2pi}displaystyle rac{dx}{a+bcos x}=displaystyle rac{2pi}{displaystyle sqrt{a^2-b^2}}$
20.: $displaystyleint_{0}^{pi/2}displaystyle rac{dx}{a+bcos x}=displaystyle rac{cos^{-1}(b/a)}{displaystyle sqrt{a^2-b^2}}$
21.: $displaystyleint_{0}^{2pi}displaystyle rac{dx}{(a+bsin x)^2}=int_{0}^{2... ...playstyle rac{dx}{(a+bcos x)^2}=displaystyle rac{2pi a}{(a^2-b^2)^{3/2}}$
22.: $displaystyleint_{0}^{2pi}displaystyle rac{dx}{1-2acos x+a^2}=displaystyle rac{2pi}{1-a^2},hspace{.2in}0<a<1$
23.: $displaystyleint_{0}^{pi}displaystyle rac{x sin x dx}{1-2acos x +a^2}=l... ...mid amid <1 piln(1+1/a) &hspace{.3in} mid amid >1 end{array} ight.$
24.: $displaystyleint_{0}^{pi}displaystyle rac{cos mx dx}{1-2acos x+a^2}=displaystyle rac{pi a^m}{1-a^2},hspace{.2in}a^2<1$ ,
m=0,1,2,...
25.: $displaystyleint_{0}^{infty}sin ax^2 dx=int_{0}^{infty}cos ax^2 dx=displaystyle rac{1}{2}displaystyle sqrt{displaystyle rac{pi}{2a}}$
26.: $displaystyleint_{0}^{infty}sin ax^n dx=displaystyle rac{1}{na^{1/n}}Gamma(1/n)sindisplaystyle rac{pi}{2n},hspace{.2in}n>1$
27.: $displaystyleint_{0}^{infty}cos ax^n dx=displaystyle rac{1}{na^{1/n}}Gamma(1/n)cosdisplaystyle rac{pi}{2n},hspace{.2in}n>1$
28.: $displaystyleint_{0}^{infty}displaystyle rac{sin x}{displaystyle sqrt{x... ...s x}{displaystyle sqrt{x}}dx=displaystyle sqrt{displaystyle rac{pi}{2}}$
29.: $displaystyleint_{0}^{infty}displaystyle rac{sin x}{x^p}dx=displaystyle rac{pi}{2Gamma (p)sin(ppi/2)},hspace{.2in}0<p<1$
30.: $displaystyleint_{0}^{infty}displaystyle rac{cos x}{x^p}dx=displaystyle rac{pi}{2Gamma (p)cos(ppi/2)},hspace{.2in}0<p<1$
31.: $displaystyleint_{0}^{infty}sin ax^2 cos 2bx dx=displaystyle rac{1}{2}d... ...}}left( cosdisplaystyle rac{b^2}{a}-sindisplaystyle rac{b^2}{a} ight)$
32.: $displaystyleint_{0}^{infty}cos ax^2cos 2bxdx=displaystyle rac{1}{2}dis... ...}}left( cosdisplaystyle rac{b^2}{a}+sindisplaystyle rac{b^2}{a} ight)$
33.: $displaystyleint_{0}^{infty}displaystyle rac{sin^3 x}{x^3}dx=displaystyle rac{3pi}{8}$
34.: $displaystyleint_{0}^{infty}displaystyle rac{sin^4 x}{x^4}dx=displaystyle rac{pi}{3}$
35.: $displaystyleint_{0}^{infty}displaystyle rac{ an x}{x}dx=displaystyle rac{pi}{2}$
36.: $displaystyleint_{0}^{pi/2}displaystyle rac{dx}{1+ an^m x}=displaystyle rac{pi}{4}$
37.: $displaystyleint_{0}^{pi/2}displaystyle rac{x}{sin x}dx=2left{ display... ...displaystyle rac{1}{5^2}-displaystyle rac{1}{7^2}+cdotcdotcdot ight}$
38.: $displaystyleint_{0}^{1}displaystyle rac{ an^{-1}x}{x}dx=displaystyle r... ...}{3^2}+displaystyle rac{1}{5^2}-displaystyle rac{1}{7^2}+ cdotcdotcdot$
39.: $displaystyleint_{0}^{1}displaystyle rac{sin^{-1}x}{x}dx=displaystyle rac{pi}{2}ln2$

DEFINITE INTEGRALS CONTAINING HYPERBOLIC FUNCTIONS

1.: $displaystyleint_{0}^{infty}displaystyle rac{sin ax}{sinh bx}dx=displaystyle rac{pi}{2b} anhdisplaystyle rac{api}{2b}$
2.: $displaystyleint_{0}^{infty}displaystyle rac{cos ax}{cosh bx}dx=displaystyle rac{pi}{2b}displaystyle rac{1}{cosh (api/2b)}$
3.: $displaystyleint_{0}^{infty}displaystyle rac{x dx}{sinh ax}=displaystyle rac{pi^2}{4a^2}$
4.: $displaystyleint_{0}^{infty}displaystyle rac{x^n dx}{sinh ax}=displaysty... ...tyle rac{1}{2^{n+1}}+displaystyle rac{1}{3^{n+1}}+cdotcdotcdot ight}$
5.: $displaystyleint_{0}^{infty}displaystyle rac{sinh ax}{e^{bx}+1}dx=displaystyle rac{pi}{2b}cscdisplaystyle rac{api}{b}-displaystyle rac{1}{2a}$
6.: $displaystyleint_{0}^{infty}displaystyle rac{sinh ax}{e^{bx}-1}dx=displaystyle rac{1}{2a}-displaystyle rac{pi}{2b}cotdisplaystyle rac{api}{b}$

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 .

Common Integrals

1.
2.
3.
4.
5.
6.
7.: $displaystyle int u^{n}du = displaystyle rac{u^{n+1}}{n+1}, n eq -1$
8.
9.: $displaystyle int e^{u}du=e^{u}$
10.: $displaystyle int a^{u}du = int e^{u ln a}du = displaystyle rac{e^{u ln a}}{ln a} = displaystyle rac{a^{u}}{ln a} , a >0, a eq 1$
11.
12.
13.
14.
15.
16.

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

27.
28.
29.: sech $u du = sin ^{-1}( anh u )$ or $2 an ^{-1}e^{u}$
30.: csch or $-coth ^{-1}e^{u}$
31.: sech $^{2} u du = anh u$
32.: csch ² u du =-coth u
33.: $displaystyle int anh ^{2} u du = u - anh u$
34.: coth ² u du = u -coth u
35.: $displaystyle intsinh ^{2} u du = displaystyle rac{sinh 2u}{4} - displaystyle rac{u}{2} = displaystyle rac{1}{2}(sinh u cosh u- u)$
36.: $displaystyle intcosh ^{2} u du = displaystyle rac{sinh 2u}{4} + displaystyle rac{u}{2} = displaystyle rac{1}{2}(sinh u cosh u+ u)$

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

COMMON SUBSTITUTIONS

1.: where
2.: where
3.: $displaystyle int Fleft( sqrt[n]{ax+b} ight) ,dx = displaystyle rac{n}{a} displaystyle int u^{n-1},F(u),du$
where
4.: $displaystyle int Fleft( displaystylesqrt{a^{2}-x^{2}} ight),dx = a,displaystyle int F(a cos u),cos u,du$
where
5.: $displaystyle int Fleft( displaystylesqrt{x^2+a^{2}} ight),dx= a,displaystyle int F(a sec u) sec ^{2} u , du$
where
6.: $displaystyle int Fleft( displaystylesqrt{x^{2}-a^{2}} ight),dx=a displaystyle int F(a an u) sec u an u,du$
where
7.: $displaystyle int F (edisplaystyle^{ax}),dx = displaystyle rac{1}{a} displaystyle intdisplaystyle rac{F(u)}{u},du$
where $u=edisplaystyle^{ax}$
8.: where
9.: $displaystyle int Fleft( sin ^{-1}displaystyle rac{x}{a} ight),dx = a,displaystyle int F(u)cos u,du$
where $u=sin ^{-1}displaystyle rac{x}{a}$

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.: $displaystyleint_{0}^{infty}displaystyle rac{e^{-ax}(1-cos x)}{x^2}dx=cot^{-1}a-displaystyle rac{a}{2}ln(a^2+1)$

Integrals with Inverse Trigonometric Functions

1.: $displaystyleintsin^{-1}displaystyle rac{x}{a}dx=xsin^{-1} displaystyle rac{x}{a}+displaystyle sqrt{a^2-x^2}$
2.: $displaystyleint xsin^{-1}displaystyle rac{x}{a}dx=left(displaystyle r... ...displaystyle rac{x}{a}+displaystyle rac{xdisplaystyle sqrt{a^2-x^2}}{4}$
3.: $displaystyleint x^2sin^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...yle rac{x}{a}+displaystyle rac{(x^2+2a^2)displaystyle sqrt{a^2-x^2)}}{9}$
4.: $displaystyleintdisplaystyle rac{sin^{-1}(x/a)}{x}dx=displaystyle rac{x... ...rac{1cdot 3cdot 5(x/a)^7}{2cdot 4cdot 6cdot 7cdot 7} + cdot cdot cdot$
5.: $displaystyleintdisplaystyle rac{sin^{-1}(x/a)}{x^2}dx=-displaystyle ra... ...rac{1}{a}lnleft(displaystyle rac{a+displaystyle sqrt{a^2-x^2}}{x} ight)$
6.: $displaystyleintleft(sin^{-1}displaystyle rac{x}{a} ight)^2 dx=xleft(s... ...a} ight)^2 -2x+2displaystyle sqrt{a^2-x^2}sin^{-1}displaystyle rac{x}{a}$
7.: $displaystyleintcos^{-1}displaystyle rac{x}{a}dx=xcos^{-1}displaystyle rac{x}{a}-displaystyle sqrt{a^2-x^2}$
8.: $displaystyleint xcos^{-1}displaystyle rac{x}{a}dx=left(displaystyle r... ...displaystyle rac{x}{a}-displaystyle rac{xdisplaystyle sqrt{a^2-x^2}}{4}$

9.: $displaystyleint x^2cos^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...tyle rac{x}{a}-displaystyle rac{(x^2+2a^2)displaystyle sqrt{a^2-x^2}}{9}$
10.: $displaystyleintdisplaystyle rac{cos^{-1}(x/a)}{x}dx=displaystyle rac{pi}{2}ln x-intdisplaystyle rac{sin^{-1}(x/a)}{x}dx$
11.: $displaystyleintdisplaystyle rac{cos^{-1}(x/a)}{x^2}dx=-displaystyle ra... ...rac{1}{a}lnleft(displaystyle rac{a+displaystyle sqrt{a^2-x^2}}{x} ight)$
12.: $displaystyleintleft(cos^{-1}displaystyle rac{x}{a} ight)^2 dx=xleft(c... ...{a} ight)^2-2x-2displaystyle sqrt{a^2-x^2}cos^{-1}displaystyle rac{x}{a}$
13.: $displaystyleint an^{-1}displaystyle rac{x}{a}dx=x an^{-1}displaystyle rac{x}{a}-displaystyle rac{a}{2}ln(x^2+a^2)$
14.: $displaystyleint x an^{-1}displaystyle rac{x}{a}dx=displaystyle rac{1}{2}(x^2+a^2) an^{-1}displaystyle rac{x}{a}-displaystyle rac{ax}{2}$
15.: $displaystyleint x^2 an^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...frac{x}{a}-displaystyle rac{ax^2}{6}+displaystyle rac{a^3}{6}ln(x^2+a^2)$

16.: $displaystyleintdisplaystyle rac{ an^{-1}(x/a)}{x}dx=displaystyle rac{x... ...playstyle rac{(x/a)^5}{5^2}-displaystyle rac{(x/a)^7}{7^2}+cdotcdotcdot$
17.: $displaystyleintdisplaystyle rac{ an^{-1}(x/a)}{x^2}dx=-displaystyle ra... ...{a}-displaystyle rac{1}{2a}lnleft(displaystyle rac{x^2+a^2}{x^2} ight)$
18.: $displaystyleintcot^{-1}displaystyle rac{x}{a}dx=xcot^{-1}displaystyle rac{x}{a}+displaystyle rac{a}{2}ln(x^2+a^2)$
19.: $displaystyleint xcot^{-1}displaystyle rac{x}{a}dx=displaystyle rac{1}{2}(x^2+a^2)cot^{-1}displaystyle rac{x}{a}+displaystyle rac{ax}{2}$
20.: $displaystyleint x^2cot^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...frac{x}{a}+displaystyle rac{ax^2}{6}-displaystyle rac{a^3}{6}ln(x^2+a^2)$
21.: $displaystyleintdisplaystyle rac{cot^{-1}(x/a)}{x}dx=displaystyle rac{pi}{2}ln x-intdisplaystyle rac{ an^{-1}(x/a)}{x}dx$

22.: $displaystyleintdisplaystyle rac{cot^{-1}(x/a)}{x^2}dx=-displaystyle ra... ...{x}+displaystyle rac{1}{2a}lnleft(displaystyle rac{x^2+a^2}{x^2} ight)$
23.: $displaystyleintsec^{-1}displaystyle rac{x}{a}dx=left{ egin{array}{ll... ...style rac{pi}{2}<sec^{-1}displaystyle rac{x}{a}<pi end{array} ight.$
24.: $displaystyleint xsec^{-1}displaystyle rac{x}{a}dx=left{ egin{array}{l... ...ystyle rac{pi}{2}<sec^{-1}displaystyle rac{x}{a}<pi end{array} ight.$
25.: $displaystyleint x^2sec^{-1}displaystyle rac{x}{a}dx=left{ egin{array}... ...ystyle rac{pi}{2}<sec^{-1}displaystyle rac{x}{a}<pi end{array} ight.$
26.: $displaystyleintdisplaystyle rac{sec^{-1}(x/a)}{x}dx=displaystyle rac{... ... rac{1cdot 3cdot 5(a/x)^7}{2cdot 4cdot 6cdot 7cdot 7} + cdotcdotcdot$
27.: $displaystyleintdisplaystyle rac{sec^{-1}(x/a)}{x^2}dx=left{ egin{arra... ...ystyle rac{pi}{2}<sec^{-1}displaystyle rac{x}{a}<pi end{array} ight.$
28.: $displaystyleintcsc^{-1}displaystyle rac{x}{a}dx=left{ displaystyleeg... ...laystyle rac{pi}{2}<csc^{-1}displaystyle rac{x}{a}<0 end{array} ight.$
29.: $displaystyleint xcsc^{-1}displaystyle rac{x}{a}dx=left{ egin{array}{l... ...laystyle rac{pi}{2}<csc^{-1}displaystyle rac{x}{a}<0 end{array} ight.$

30.: $displaystyleint x^2csc^{-1}displaystyle rac{x}{a}dx=left{ egin{array}... ...laystyle rac{pi}{2}<csc^{-1}displaystyle rac{x}{a}<0 end{array} ight.$
31.: $displaystyleintdisplaystyle rac{csc^{-1}(x/a)}{x}dx=-left(displaystyle ... ...{1cdot 3cdot 5(a/x)^7}{2cdot 4cdot 6cdot 7cdot 7}+cdotcdotcdot ight)$
32.: $displaystyleintdisplaystyle rac{csc^{-1}(x/a)}{x^2}dx=left{ egin{arra... ...laystyle rac{pi}{2}<csc^{-1}displaystyle rac{x}{a}<0 end{array} ight.$
33.: $displaystyleint x^msin^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...e rac{1}{m+1}intdisplaystyle rac{x^{m+1}}{displaystyle sqrt{a^2-x^2}}dx$
34.: $displaystyleint x^mcos^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...e rac{1}{m+1}intdisplaystyle rac{x^{m+1}}{displaystyle sqrt{a^2-x^2}}dx$
35.: $displaystyleint x^m an^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...ac{x}{a}-displaystyle rac{a}{m+1}intdisplaystyle rac{x^{m+1}}{x^2+a^2}dx$
36.: $displaystyleint x^mcot^{-1}displaystyle rac{x}{a}dx=displaystyle rac{x... ...ac{x}{a}+displaystyle rac{a}{m+1}intdisplaystyle rac{x^{m+1}}{x^2+a^2}dx$
37.: $displaystyleint x^msec^{-1}displaystyle rac{x}{a}=left { egin{array}{... ...aystyle rac{pi}{2}<sec^{-1}displaystyle rac{x}{a}<pi end{array} ight.$
38.: $displaystyleint x^mcsc^{-1}displaystyle rac{x}{a}dx=left{ egin{array}... ...laystyle rac{pi}{2}<csc^{-1}displaystyle rac{x}{a}<0 end{array} ight.$