int frac{(sin^{-1} x)^{frac{3}{2}}}{sqrt{1-x^ | vedclass

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Question

(A) Let $I = \int \frac{(\sin^{-1} x)^{\frac{3}{2}}}{\sqrt{1-x^2}} dx$. Substitute $\sin^{-1} x = t$. Then,differentiating both sides with respect to

Vedclass · Accepted Answer

$\frac{2}{5}(\sin^{-1} x)^{\frac{5}{2}} + c$

Vedclass · Answer

(A) Let $I = \int \frac{(\sin^{-1} x)^{\frac{3}{2}}}{\sqrt{1-x^2}} dx$. Substitute $\sin^{-1} x = t$. Then,differentiating both sides with respect to $x$,we get $\frac{1}{\sqrt{1-x^2}} dx = dt$. Substituting these into the integral,we have: $I = \int t^{\frac{3}{2}} dt$. Using the power rule for integration $\int t^n dt = \frac{t^{n+1}}{n+1} + c$: $I = \frac{t^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} + c = \frac{t^{\frac{5}{2}}}{\frac{5}{2}} + c = \frac{2}{5} t^{\frac{5}{2}} + c$. Substituting back $t = \sin^{-1} x$,we get: $I = \frac{2}{5}(\sin^{-1} x)^{\frac{5}{2}} + c$.

$\int \frac{(\sin^{-1} x)^{\frac{3}{2}}}{\sqrt{1-x^2}} dx =$

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