∫10 x7√1−x4dx is equal to

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Question

\int_{0}^{1} \frac{x^{7}}{\sqrt{1 - x^{4}}} d x

is equal to

Options:

A . 1

B . 13

C . 23

D . π3

Answer: Option B
:
B

I = \int_{0}^{1} \frac{x^{7}}{\sqrt{1 - x^{4}}} d x = \int_{0}^{1} \frac{x^{6} x d x}{\sqrt{1 - x^{4}}}

P u t x^{2} = s i n θ \Rightarrow 2 x d x = c o s θ d θ

I = \frac{1}{2}_{0}^{\frac{π}{2}} \frac{s i n^{3} θ . c o s θ d θ}{c o s θ} = \frac{1}{2} \int_{0}^{\frac{1}{2}} s i n^{3} θ d θ

= \frac{1}{2} \frac{T 2 T (\frac{1}{2})}{2. T (\frac{1}{2})} = \frac{T (\frac{1}{2})}{4. \frac{3}{2} . \frac{1}{2} . T (\frac{1}{2})} = \frac{1}{3}

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