Question

Integration of $\mathit{\int x sin^{-1}x dx}$

Answer 1

I hope you will be able to understand

Answer 2

$Integrating by parts</p><p></p><p>=sin−1(x)∫x⋅dx−∫[ddx[sin−1(x)]∫xdx]dx=sin−1⁡(x)∫x⋅dx−∫[ddx[sin−1⁡(x)]∫xdx]dx</p><p></p><p>=x2sin−1(x)2−12∫(x21−x2−−−−−√)dx=x2sin−1⁡(x)2−12∫(x21−x2)dx..(1)</p><p></p><p>Now we have to solve</p><p></p><p>I=∫(x21−x2−−−−−√)dxI=∫(x21−x2)dx</p><p></p><p>Substitution 1−x2=t21−x2=t2</p><p></p><p>−x⋅dx=t⋅dt−x⋅dx=t⋅dt</p><p></p><p>dx=−t⋅dtxdx=−t⋅dtx</p><p></p><p>I=−∫x2t⋅t⋅dtxI=−∫x2t⋅t⋅dtx</p><p></p><p>=−x⋅dt=−x⋅dt</p><p></p><p>=−1−t2−−−−−√⋅dt=−1−t2⋅dt</p><p></p><p>=−[t21−t2−−−−−√+12sin−1(t)]=−[t21−t2+12sin−1⁡(t)]</p><p></p><p>Substituting t2=1−x2t2=1−x2</p><p></p><p>=−[1−x2−−−−−√2x+12sin−1(1−x2−−−−−√)]=−[1−x22x+12sin−1⁡(1−x2)]</p><p></p><p>Putting II in equation (1)(1)</p><p></p><p>∫xsin−1(x)dx=∫xsin−1⁡(x)dx=</p><p></p><p>12[1−x2−−−−−√2x+12sin−1(1−x2−−−−−√)]+x2sin−1(x)2+constant12[1−x22x+12sin−1⁡(1−x2)]+x2sin−1⁡(x)2+constant</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>$