Let $\frac{d}{dx}\ F(x) = \left( \frac{e^{\sin x}}{x} \right),x 0$, If $\int_{1}^{4}\frac{3}{x}\ e^{\sin x^{3}}\ dx = F(k) - F(1)$, then one of the possible values of $k$, is

Question

Let $\frac{d}{dx}\ F(x) = \left( \frac{e^{\sin x}}{x} \right),x > 0$, If $\int_{1}^{4}\frac{3}{x}\ e^{\sin x^{3}}\ dx = F(k) - F(1)$, then one of the possible values of $k$, is

Dr. Abubacker · Accepted Answer

(d)
Given, $\frac{d}{dx}F(x) = \frac{e^{\sin x}}{x},\ x > 0$
On integrating both sides, we get
$F(x) = \int_{}^{}{\frac{e^{\sin x}}{x}dx}$ …(i)
Also $\int_{1}^{4}\frac{3}{x}e^{\sin x^{3}}dx = \int_{1}^{4}{\frac{3x^{2}}{x^{3}}.e^{\sin x^{3}}dx = F(k) - F(1)}$
Let $x^{3} = z \Rightarrow 3x^{2}dx = dz$
$	herefore\ \ \int_{1}^{64}\frac{e^{\sin z}}{z}dz = F(k) - F(1)$
$\Rightarrow \lbrack F{(z)brack}_{1}^{64} = F(k) - F(1)$ [from Eq.(i)]
$\Rightarrow F(64) - F(1) = F(k) - F(1)$
$\Rightarrow k = 64$

Q. Let $\frac{d}{dx}\ F(x) = \left( \frac{e^{\sin x}}{x} \right),x > 0$, If $\int_{1}^{4}\frac{3}{x}\ e^{\sin x^{3}}\ dx = F(k) - F(1)$, then one of the possible values of $k$, is

Explanation

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