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Table of Integrals - Forms Involving Inverse Trigonometric Functions

The integrals below involve inverse trigonometric functions.

1)  sin-1(xa) dx=xsin-1(xa)+a2-x2

2)  xsin-1(xa) dx=(x22-a24)sin-1(xa)+xa2-x24

3)  x2sin-1(xa) dx=x33sin-1(xa)+(x2+2a2)a2-x29

4)  sin-1(xa)x dx=xa+(xa)3233+13(xa)52455+135(xa)724677+...

5)  sin-1(xa)x2 dx=-sin-1(xa)x-1aln(a+a2-x2x)

6)  [sin-1(xa)]2 dx=x[sin-1(xa)]2-2x+2a2-x2sin-1(xa)

7)  cos-1(xa) dx=xcos-1(xa)-a2-x2

8)  xcos-1(xa) dx=(x22-a24)cos-1(xa)-xa2-x24

9)  x2cos-1(xa) dx=x33cos-1(xa)-(x2+2a2)a2-x29

10)  cos-1(xa)x dx=π2lnx- sin-1(xa)x dx

                    **[See integral #4 in this table]

11)  cos-1(xa)x2 dx=-cos-1(xa)x+1aln(a+a2-x2x)

12)  [cos-1(xa)]2 dx=x[cos-1(xa)]2-2x-2a2-x2cos-1(xa)

13)  tan-1(xa) dx=xtan-1(xa)-a2ln(x2+a2)

14)  xtan-1(xa) dx=12(x2+a2)tan-1(xa)-ax2

15)  x2tan-1(xa) dx=x33tan-1(xa)-ax26+a36ln(x2+a2)

16)  tan-1(xa)x dx=xa-(xa)332+(xa)552-(xa)772+...

17)  tan-1(xa)x2 dx=-1xtan-1(xa)-12aln(x2+a2x2)

18)  cot-1(xa) dx=xcot-1(xa)+a2ln(x2+a2)

19)  xcot-1(xa) dx=12(x2+a2)cot-1(xa)+ax2

20)  x2cot-1(xa) dx=x33cot-1(xa)+ax26-a36ln(x2+a2)

21)  cot-1(xa)x dx=π2lnx- tan-1(xa)x dx

                    **[See integral #16 in this table]

22)  cot-1(xa)x2 dx=-cot-1(xa)x+12aln(x2+a2x2)

23)  sec-1(xa) dx=xsec-1(xa)-aln(x+x2-a2).....For 0<sec-1(xa)<π2

                        OR =xsec-1(xa)+aln(x+x2-a2).....For π2<sec-1(xa)<π

24)  xsec-1(xa) dx=x22sec-1(xa)-ax2-a22.....For 0<sec-1(xa)<π2

                             OR =x22sec-1(xa)+ax2-a22.....For π2<sec-1(xa)<π

25)  x2sec-1(xa) dx=x33sec-1(xa)-axx2-a26-a36ln(x+x2-a2).....For 0<sec-1(xa)<π2

                               OR =x33sec-1(xa)+axx2-a26+a36ln(x+x2-a2).....For π2<sec-1(xa)<π

26)  sec-1(xa)x dx=π2lnx+ax+(ax)3233+13(ax)52455+135(ax)724677+...

27)  sec-1(xa)x2 dx=-sec-1(xa)x+x2-a2ax.....For 0<sec-1(xa)<π2

                        OR =-sec-1(xa)x-x2-a2ax.....For π2<sec-1(xa)<π

28)  csc-1(xa) dx=xcsc-1(xa)+aln(x+x2-a2).....For 0<csc-1(xa)<π2

                        OR =xcsc-1(xa)-aln(x+x2-a2).....For -π2<csc-1(xa)<0

29)  xcsc-1(xa) dx=x22csc-1(xa)+ax2-a22.....For 0<csc-1(xa)<π2

                             OR =x22csc-1(xa)-ax2-a22.....For -π2<csc-1(xa)<0

30)  x2csc-1(xa) dx=x33csc-1(xa)+axx2-a26+a36ln(x+x2-a2).....For 0<csc-1(xa)<π2

                            OR =x33csc-1(xa)-axx2-a26-a36ln(x+x2-a2).....For -π2<csc-1(xa)<0

31)  csc-1(xa)x dx=-(ax+(ax)3233+13(ax)52455+135(ax)724677+...)

32)  csc-1(xa)x2 dx=-csc-1(xa)x-x2-a2ax.....For 0<csc-1(xa)<π2

                        OR =-csc-1(xa)x+x2-a2ax.....For -π2<csc-1(xa)<0

33)  xmsin-1(xa) dx=xm+1m+1sin-1(xa)-1m+1 xm+1a2-x2 dx

34)  xmcos-1(xa) dx=xm+1m+1cos-1(xa)+1m+1 xm+1a2-x2 dx

35)  xmtan-1(xa) dx=xm+1m+1tan-1(xa)-am+1 xm+1x2+a2 dx

36)  xmcot-1(xa) dx=xm+1m+1cot-1(xa)+am+1 xm+1x2+a2 dx

37)  xmsec-1(xa) dx=xm+1sec-1(xa)m+1-am+1 xmx2-a2 dx.....For 0<sec-1(xa)<π2

                             OR =xm+1sec-1(xa)m+1+am+1 xmx2-a2 dx.....For π2<sec-1(xa)<π

38)  xmcsc-1(xa) dx=xm+1csc-1(xa)m+1+am+1 xmx2-a2 dx.....For 0<csc-1(xa)<π2

                             OR =xm+1csc-1(xa)m+1-am+1 xmx2-a2 dx.....For -π2<csc-1(xa)<0