Question

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

• A. 15
• B. 16
• C. 63
• D. 64

Answer 1

Answer: (D)

64

Answer 2

$\frac{d}{dx}$F(x) = $\frac{e^{\sin x}}{x}$, x > 0

On integrating both sides of above equation, we get

F(x) = $\int_{}^{}\frac{e^{\sin x}}{x}$dx …(1)

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 x³ = z Ž 3x² dx = dz

Ž $\int_{1}^{64}\frac{e^{\sin z}}{z}$dz = F(k) – F(1)

Using equation (1), we get

[F(z)$\rbrack_{1}^{64}$ = F(k) – F(1)

Ž F(64) – F(1) = F(k) – F(1)

Ž k = 64.

Hence (4) is the correct answer.