Indefinite Integration 1 Question 90
91. The value of $\int _0^{1} 4 x^{3} \frac{d^{2}}{d x^{2}}\left(1-x^{2}\right)^{5} d x$ is
(2014 Adv.)
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Answer:
Correct Answer: 91. (2)
Solution:
- PLAN Integration by parts
$\int f(x) g(x) d x=f(x) \int g(x) d x-\int \frac{d}{d x}[f(x)] \int g(x) d x d x$
Given, $I=\int _0^{1} 4 x^{3} \frac{d^{2}}{d x^{2}}\left(1-x^{2}\right)^{5} d x$
$$ =4 x^{3} \frac{d}{d x}\left(1-x^{2}\right)^{5}{ } _0^{1}-\int _0^{1} 12 x^{2} \frac{d}{d x}\left(1-x^{2}\right)^{5} d x $$
$$ \begin{aligned} & =4 x^{3} \times 5\left(1-x^{2}\right)^{4}(-2 x) \\ & \quad-12\left[x^{2}\left(1-x^{2}\right)^{5}\right] _0^{1}-\int _0^{1} 2 x\left(1-x^{2}\right)^{5} d x \\ & =0-0-12(0-0)+12 \int _0^{1} 2 x\left(1-x^{2}\right)^{5} d x \\ & =12 \times-\frac{\left(1-x^{2}\right)^{6}}{6}{ } _0^{1}=120+\frac{1}{6}=2 \end{aligned} $$
