The integrals below involve inverse trigonometric functions.
1) ∫ sin-1(xa) dx=x⋅sin-1(xa)+√a2-x2
2) ∫ x⋅sin-1(xa) dx=(x22-a24)sin-1(xa)+x√a2-x24
3) ∫ x2sin-1(xa) dx=x33sin-1(xa)+(x2+2a2)√a2-x29
4) ∫ sin-1(xa)x dx=xa+(xa)32⋅3⋅3+1⋅3(xa)52⋅4⋅5⋅5+1⋅3⋅5(xa)72⋅4⋅6⋅7⋅7+...
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+2√a2-x2⋅sin-1(xa)
7) ∫ cos-1(xa) dx=x⋅cos-1(xa)-√a2-x2
8) ∫ x⋅cos-1(xa) dx=(x22-a24)cos-1(xa)-x√a2-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-2√a2-x2⋅cos-1(xa)
13) ∫ tan-1(xa) dx=x⋅tan-1(xa)-a2ln(x2+a2)
14) ∫ x⋅tan-1(xa) dx=12(x2+a2)tan-1(xa)-ax2
15) ∫ x2⋅tan-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=x⋅cot-1(xa)+a2ln(x2+a2)
19) ∫ x⋅cot-1(xa) dx=12(x2+a2)cot-1(xa)+ax2
20) ∫ x2⋅cot-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=x⋅sec-1(xa)-aln(x+√x2-a2).....For 0<sec-1(xa)<π2
OR =x⋅sec-1(xa)+aln(x+√x2-a2).....For π2<sec-1(xa)<π
24) ∫ x⋅sec-1(xa) dx=x22sec-1(xa)-a√x2-a22.....For 0<sec-1(xa)<π2
OR =x22sec-1(xa)+a√x2-a22.....For π2<sec-1(xa)<π
25) ∫ x2⋅sec-1(xa) dx=x33sec-1(xa)-ax√x2-a26-a36ln(x+√x2-a2).....For 0<sec-1(xa)<π2
OR =x33sec-1(xa)+ax√x2-a26+a36ln(x+√x2-a2).....For π2<sec-1(xa)<π
26) ∫ sec-1(xa)x dx=π2lnx+ax+(ax)32⋅3⋅3+1⋅3(ax)52⋅4⋅5⋅5+1⋅3⋅5(ax)72⋅4⋅6⋅7⋅7+...
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=x⋅csc-1(xa)+aln(x+√x2-a2).....For 0<csc-1(xa)<π2
OR =x⋅csc-1(xa)-aln(x+√x2-a2).....For -π2<csc-1(xa)<0
29) ∫ x⋅csc-1(xa) dx=x22csc-1(xa)+a√x2-a22.....For 0<csc-1(xa)<π2
OR =x22csc-1(xa)-a√x2-a22.....For -π2<csc-1(xa)<0
30) ∫ x2csc-1(xa) dx=x33csc-1(xa)+ax√x2-a26+a36ln(x+√x2-a2).....For 0<csc-1(xa)<π2
OR =x33csc-1(xa)-ax√x2-a26-a36ln(x+√x2-a2).....For -π2<csc-1(xa)<0
31) ∫ csc-1(xa)x dx=-(ax+(ax)32⋅3⋅3+1⋅3(ax)52⋅4⋅5⋅5+1⋅3⋅5(ax)72⋅4⋅6⋅7⋅7+...)
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+1√a2-x2 dx
34) ∫ xmcos-1(xa) dx=xm+1m+1cos-1(xa)+1m+1∫ xm+1√a2-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∫ xm√x2-a2 dx.....For 0<sec-1(xa)<π2
OR =xm+1sec-1(xa)m+1+am+1∫ xm√x2-a2 dx.....For π2<sec-1(xa)<π
38) ∫ xmcsc-1(xa) dx=xm+1csc-1(xa)m+1+am+1∫ xm√x2-a2 dx.....For 0<csc-1(xa)<π2
OR =xm+1csc-1(xa)m+1-am+1∫ xm√x2-a2 dx.....For -π2<csc-1(xa)<0