The value(s) of _0^1 x^4(1-x)^4 1+x^2 d x is (are) - Vidyadip

MCQ

The value(s) of $\int_0^1 \frac{x^4(1-x)^4}{1+x^2} d x$ is (are)

✓
$\frac{22}{7}-\pi$
B
$\frac{2}{105}$
C
$0$
D
$\frac{71}{15}-\frac{3 \pi}{2}$

✓

Answer

Correct option: A.

$\frac{22}{7}-\pi$

a
$ \int_0^1 \frac{x^4(1-x)^4}{1+x^2} d x $

$ =\int_0^1\left(x^6-4 x^5+5 x^4-4 x^2+4-\frac{4}{1+x^2}\right) d x $

$ =\left[\frac{x^7}{7}-\frac{2 x^6}{3}+x^5-\frac{4 x^3}{3}+4 x\right]_0^1-\pi $

$ =\frac{1}{7}-\frac{2}{3}+1-\frac{4}{3}+4-\pi=\frac{22}{7}-\pi$

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