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

If φ(x)=∫1/xxsin⁡(t2)dt,\varphi(x) = \int_{1/x}^{\sqrt{x}}{\sin(t^{2})dt,} then φ′(1)=\varphi^{'}(1) =

A

sin⁡1\sin 1

B

2sin⁡12\sin 1

C

32sin⁡1\frac{3}{2}\sin 1

D

None of these

Answer

32sin⁡1\frac{3}{2}\sin 1

Explanation

Solution

φ′(x)=sin⁡xddxx−sin⁡1x2ddx(1x)\varphi'(x) = \sin x\frac{d}{dx}\sqrt{x} - \sin\frac{1}{x^{2}}\frac{d}{dx}\left( \frac{1}{x} \right)

=sin⁡x.12x+1x2sin⁡1x2= \sin x.\frac{1}{2\sqrt{x}} + \frac{1}{x^{2}}\sin\frac{1}{x^{2}}

⇒ φ′(1)=12sin⁡1+sin⁡1=32sin⁡1\varphi'(1) = \frac{1}{2}\sin 1 + \sin 1 = \frac{3}{2}\sin 1.