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Evaluate the following problems using second fundamental theorem: $\int\limits_{0}^{\pi/2}\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 $\int \limits_0^1 \large\frac{dx}{\sqrt {4-x^2}}$
Step 1:
$\int \limits_0^1 \large\frac{dx}{\sqrt {4-x^2}}=\int \limits_0^1 \large\frac{dx}{\sqrt {2^2-x^2}}$
Step 2:
$\qquad=\sin ^{-1} \large\frac{x}{2} \bigg]_0^1$
$\qquad=\sin ^{-1} \large\frac{1}{2}$$-\sin ^{-1}0$
$\qquad=\large\frac{\pi}{6}$
answered Aug 14, 2013 by meena.p
 

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