If (x) = _ 1/x ^ x ( t^2 ) ,dt, then '(1) = - Vidyadip

MCQ

If $\varphi (x) = \int_{1/x}^{\sqrt x } {\sin ({t^2})\,dt,} $ then $\phi '(1) = $

A
$\sin 1$
B
$2\sin 1$
✓
$\frac{3}{2}\sin 1$
D
None of these

✓

Answer

Correct option: C.

$\frac{3}{2}\sin 1$

c
(c) $\phi '(x) = \sin x\frac{d}{{dx}}\sqrt x - \sin \frac{1}{{{x^2}}}\frac{d}{{dx}}\left( {\frac{1}{x}} \right)$

$ = \sin x.\frac{1}{{2\sqrt x }} + \frac{1}{{{x^2}}}\sin \frac{1}{{{x^2}}}$

==> $\phi '(1) = \frac{1}{2}\sin 1 + \sin 1 = \frac{3}{2}\sin 1$.

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