Let  ddx(F(x))=esin⁡xx,x>0 if ∫14 2esin⁡x2xdx=F(k)−F(1) ), then one of the possible values of k is,

Let  $\frac{d}{dx}\left(F\left(x\right)\right)=\frac{{e}^{\mathrm{sin}x}}{x},x>0$ if ${\int }_{1}^{4} \frac{2{e}^{\mathrm{sin}{x}^{2}}}{x}dx=F\left(k\right)-F\left(1\right)$ ), then one of the possible values of k is,

1. A

4

2. B

8

3. C

16

4. D

32

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Solution:

We have,

$\frac{d}{dx}\left(F\left(x\right)\right)=\frac{{e}^{\mathrm{sin}x}}{x}⇒\int \frac{{e}^{\mathrm{sin}x}}{x}dx=F\left(x\right)$               …(i)

Now,

${\int }_{1}^{4}\frac{{e}^{\mathrm{sin}{x}^{2}}}{x}dx={\int }_{1}^{4} \frac{{e}^{\mathrm{sin}{x}^{2}}}{{x}^{2}}\cdot d\left({x}^{2}\right)={\int }_{1}^{16} \frac{{e}^{\mathrm{sin}t}}{t}dt$, where $t={x}^{2}$

${\int }_{1}^{4}2\frac{{e}^{\mathrm{sin}{x}^{2}}}{x}dx=\left[F\left(t\right){\right]}^{16}=F\left(16\right)-F\left(1\right)$      [using (i)]

hence , $k=16$

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