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Evaluate the following problems using second fundamental theorem: $\int\limits_{0}^{1}\large\frac{(\sin^{-1}x)^{3}}{\sqrt{1-x^{2}}}$$dx$

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Toolbox:
  • If $F(x)=\int \limits_a^x f(t)dt $ then $\int \limits_a^b f(x) dx=F(b)-F(a)$
Given $\int\limits_{0}^{1}\large\frac{(\sin^{-1}x)^{3}}{\sqrt{1-x^{2}}}$$dx$
Step 1:
$I=\int\limits_{0}^{1}\large\frac{(\sin^{-1}x)^{3}}{\sqrt{1-x^{2}}}$$dx$
Let $t=\sin ^{-1} x$
$dt=\large\frac{1}{\sqrt {1-x^2}}$$dx$
When $x=0\quad t=0$
When $x=1\quad t=\large\frac{\pi}{2}$
Step 2:
$I=\int \limits_0^{\large\frac{\pi}{2}} t^{-3} \;dt$
$\quad=\large\frac{t^4}{4} \bigg]_0^ {\large\frac{\pi}{2}}$
$\quad=\large\frac{\pi^4}{64}$
answered Aug 14, 2013 by meena.p
 

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