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`