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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
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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