_0^ 4 ,/ , ( 3 , x^2 , ,. , , ,1 x , , - , ,x , ,. , , ,1 x ) dx … - Vidyadip

MCQ

$\int\limits_0^{4\,/\,\pi } {} $ $\left( {3\,{x^2}\,\,.\,\,\sin \,\frac{1}{x}\,\, - \,\,x\,\,.\,\,\cos \,\frac{1}{x}} \right)$ $dx$ has the value :

A
$\frac{{8\,\sqrt 2 }}{{{\pi ^3}}}$
B
$\frac{{24\,\sqrt 2 }}{{{\pi ^3}}}$
✓
$\frac{{32\,\sqrt 2 }}{{{\pi ^3}}}$
D
None

✓

Answer

Correct option: C.

$\frac{{32\,\sqrt 2 }}{{{\pi ^3}}}$

c
Consider $\int\limits_0^{\frac{4}{\pi }} {3{x^2}\,\sin \frac{1}{x}dx} $ and $IBP $ taking as the $1^{st}$ and $3x^2$ as the $2^{nd} $ function.

Two integrals cancel
$\int {\,\,\left( {3{x^2}\underbrace {\sin \frac{1}{x}}_I - x\cos \frac{1}{x}} \right)\,dx} $

$=\sin \frac{1}{x}\,\cdot\,{x^3}$$-\int {\cos \frac{1}{x}\left( { - \frac{1}{{{x^2}}}} \right){x^3}\,dx} $$-\int {x{{\cos }^2}\frac{1}{x}\,dx} $

=$\left. {{x^3}\,\cdot\,\sin \frac{1}{x}} \right|_{\,0}^{\frac{4}{\pi }}$

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