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Table of Integrals - Forms Involving Inverse Trigonometric Functions

The integrals below involve inverse trigonometric functions.

1) $int sin^-1 (x/a) dx = x*sin^-1 (x/a)+sqrt(a^2-x^2)$

2) $int x*sin^-1(x/a) dx = (x^2/2-a^2/4)sin^-1(x/a)+(xsqrt(a^2-x^2))/4$

3) $int x^2 sin^-1(x/a) dx = x^3/3sin^-1(x/a)+((x^2+2a^2)sqrt(a^2-x^2))/9$

4) $int (sin^-1(x/a))/x dx = x/a+(x/a)^3/(2*3*3)+(1*3(x/a)^5)/(2*4*5*5)+(1*3*5(x/a)^7)/(2*4*6*7*7)+...$

5) $int (sin^-1(x/a))/x^2 dx = -(sin^-1(x/a))/x-1/a ln((a+sqrt(a^2-x^2))/x)$

6) $int [sin^-1(x/a)]^2 dx = x[sin^-1(x/a)]^2-2x+2sqrt(a^2-x^2)*sin^-1(x/a)$

7) $int cos^-1(x/a) dx = x*cos^-1(x/a)-sqrt(a^2-x^2)$

8) $int x*cos^-1(x/a) dx = (x^2/2-a^2/4)cos^-1(x/a)-(xsqrt(a^2-x^2))/4$

9) $int x^2 cos^-1(x/a) dx = x^3/3 cos^-1(x/a)-((x^2+2a^2)sqrt(a^2-x^2))/9$

10) $int (cos^-1(x/a))/x dx = pi/2 lnx-int (sin^-1(x/a))/x dx$

**[See integral #4 in this table]

11) $int (cos^-1(x/a))/x^2 dx = -(cos^-1(x/a))/x+1/aln((a+sqrt(a^2-x^2))/x)$

12) $int [cos^-1(x/a)]^2 dx = x[cos^-1(x/a)]^2-2x-2sqrt(a^2-x^2) *cos^-1(x/a)$

13) $int tan^-1(x/a) dx = x*tan^-1(x/a)-a/2ln(x^2+a^2)$

14) $int x*tan^-1(x/a) dx = 1/2(x^2+a^2)tan^-1(x/a)-(ax)/2$

15) $int x^2*tan^-1(x/a) dx = x^3/3tan^-1(x/a)-(ax^2)/6+a^3/6ln(x^2+a^2)$

16) $int (tan^-1(x/a))/x dx = x/a-(x/a)^3/3^2+(x/a)^5/5^2-(x/a)^7/7^2+...$

17) $int (tan^-1(x/a))/x^2 dx = -1/xtan^-1(x/a)-1/(2a)ln((x^2+a^2)/x^2)$

18) $int cot^-1(x/a) dx = x*cot^-1(x/a)+a/2ln(x^2+a^2)$

19) $int x*cot^-1(x/a) dx = 1/2(x^2+a^2)cot^-1(x/a)+(ax)/2$

20) $int x^2 *cot^-1(x/a) dx = x^3/3cot^-1(x/a)+(ax^2)/6-a^3/6ln(x^2+a^2)$

21) $int (cot^-1(x/a))/x dx = pi/2lnx-int (tan^-1(x/a))/x dx$

**[See integral #16 in this table]

22) $int (cot^-1(x/a))/x^2 dx = -(cot^-1(x/a))/x+1/(2a)ln((x^2+a^2)/x^2)$

23) $int sec^-1(x/a) dx = x*sec^-1(x/a)-a ln(x+sqrt(x^2-a^2))$ .....For $0 < sec^-1(x/a) < pi/2$

OR $= x*sec^-1(x/a)+a ln(x+sqrt(x^2-a^2))$ .....For $pi/2 < sec^-1(x/a) < pi$

24) $int x*sec^-1(x/a) dx = x^2/2sec^-1(x/a)-(asqrt(x^2-a^2))/2$ .....For $0 < sec^-1(x/a) < pi/2$

OR $= x^2/2sec^-1(x/a)+(asqrt(x^2-a^2))/2$ .....For $pi/2 < sec^-1(x/a) < pi$

25) $int x^2*sec^-1(x/a) dx = x^3/3sec^-1(x/a)-(axsqrt(x^2-a^2))/6-a^3/6ln(x+sqrt(x^2-a^2))$ .....For $0 < sec^-1(x/a) < pi/2$

OR $= x^3/3sec^-1(x/a)+(axsqrt(x^2-a^2))/6+a^3/6ln(x+sqrt(x^2-a^2))$ .....For $pi/2 < sec^-1(x/a) < pi$

26) $int (sec^-1(x/a))/x dx = pi/2ln x+a/x+(a/x)^3/(2*3*3)+(1*3(a/x)^5)/(2*4*5*5)+(1*3*5(a/x)^7)/(2*4*6*7*7)+...$

27) $int (sec^-1(x/a))/x^2 dx = -(sec^-1(x/a))/x+sqrt(x^2-a^2)/(ax)$ .....For $0 < sec^-1(x/a) < pi/2$

OR $= -(sec^-1(x/a))/x-sqrt(x^2-a^2)/(ax)$ .....For $pi/2 < sec^-1(x/a) < pi$

28) $int csc^-1(x/a) dx = x*csc^-1(x/a)+a ln(x+sqrt(x^2-a^2))$ .....For $0 < csc^-1(x/a) < pi/2$

OR $= x*csc^-1(x/a)-a ln(x+sqrt(x^2-a^2))$ .....For $-pi/2 < csc^-1(x/a) < 0$

29) $int x*csc^-1(x/a) dx = x^2/2 csc^-1(x/a)+(asqrt(x^2-a^2))/2$ .....For $0 < csc^-1(x/a) < pi/2$

OR $= x^2/2 csc^-1(x/a)-(asqrt(x^2-a^2))/2$ .....For $-pi/2 < csc^-1(x/a) < 0$

30) $int x^2 csc^-1(x/a) dx = x^3/3csc^-1(x/a)+(axsqrt(x^2-a^2))/6+a^3/6ln(x+sqrt(x^2-a^2))$ .....For $0 < csc^-1(x/a) < pi/2$

OR $= x^3/3csc^-1(x/a)-(axsqrt(x^2-a^2))/6-a^3/6ln(x+sqrt(x^2-a^2))$ .....For $-pi/2 < csc^-1(x/a) < 0$

31) $int (csc^-1(x/a))/x dx = -(a/x+(a/x)^3/(2*3*3)+(1*3(a/x)^5)/(2*4*5*5)+(1*3*5(a/x)^7)/(2*4*6*7*7)+...)$

32) $int (csc^-1(x/a))/x^2 dx = -(csc^-1(x/a))/x-sqrt(x^2-a^2)/(ax)$ .....For $0 < csc^-1(x/a) < pi/2$

OR $= -(csc^-1(x/a))/x+sqrt(x^2-a^2)/(ax)$ .....For $-pi/2 < csc^-1(x/a) < 0$

33) $int x^m sin^-1(x/a) dx = x^(m+1)/(m+1)sin^-1(x/a)-1/(m+1) int x^(m+1)/sqrt(a^2-x^2) dx$

34) $int x^mcos^-1(x/a) dx = x^(m+1)/(m+1)cos^-1(x/a)+1/(m+1)int x^(m+1)/sqrt(a^2-x^2) dx$

35) $int x^mtan^-1(x/a) dx = x^(m+1)/(m+1)tan^-1(x/a)-a/(m+1)int x^(m+1)/(x^2+a^2) dx$

36) $int x^mcot^-1(x/a) dx = x^(m+1)/(m+1)cot^-1(x/a)+a/(m+1)int x^(m+1)/(x^2+a^2) dx$

37) $int x^msec^-1(x/a) dx = (x^(m+1)sec^-1(x/a))/(m+1)-a/(m+1)int x^m/sqrt(x^2-a^2) dx$ .....For $0 < sec^-1(x/a) < pi/2$

OR $= (x^(m+1)sec^-1(x/a))/(m+1)+a/(m+1)int x^m/sqrt(x^2-a^2) dx$ .....For $pi/2 < sec^-1(x/a) < pi$

38) $int x^mcsc^-1(x/a) dx = (x^(m+1)csc^-1(x/a))/(m+1)+a/(m+1)int x^m/sqrt(x^2-a^2) dx$ .....For $0 < csc^-1(x/a) < pi/2$

OR $= (x^(m+1)csc^-1(x/a))/(m+1)-a/(m+1)int x^m/sqrt(x^2-a^2) dx$ .....For $-pi/2 < csc^-1(x/a) < 0$