Definite Integration Question 89
Question 89
- 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.)
Show Answer
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, $I=\int_{0}^{1} 4 x^{3} \frac{d^{2}}{d x^{2}}\left(1-x^{2}\right)^{5} d x$
$$ =4 x^{3} \frac{d}{d x}\left(1-x^{2}\right)^{5}{ }{0}^{1}-\int{0}^{1} 12 x^{2} \frac{d}{d x}\left(1-x^{2}\right)^{5} d x $$
$$ \begin{aligned} & =4 x^{3} \times 5\left(1-x^{2}\right)^{4}(-2 x) \ & \quad-12\left[x^{2}\left(1-x^{2}\right)^{5}\right]{0}^{1}-\int{0}^{1} 2 x\left(1-x^{2}\right)^{5} d x \ & =0-0-12(0-0)+12 \int_{0}^{1} 2 x\left(1-x^{2}\right)^{5} d x \ & =12 \times-\frac{\left(1-x^{2}\right)^{6}}{6}{ }_{0}^{1}=120+\frac{1}{6}=2 \end{aligned} $$
