Question

Let $\frac{d}{dx}F(x) = \frac{e^{\sin x}}{x}$, x > 0. If $\int_{1}^{4}{\frac{2e^{\sin x^{2}}}{x}dx}$= f(k) – f(1) then one of the possible value of k is:

• A. 2
• B. 4
• C. 8
• D. 16

Answer 1

Answer: (D)

16

Answer 2

Let I = $\int_{1}^{4}{\frac{2e^{\sin x^{2}}}{x}dx}$= $\int_{1}^{4}\frac{2xe^{\sin x^{2}}}{x^{2}}$

Put x² = y, then 2x dx = dy

$\begin{matrix} x & 1 & 4 \\ y & 1 & 16 \end{matrix}$

\ I = $\int_{1}^{16}\frac{e^{\sin y}}{y}$dy = |F(y)|$\int_{1}^{16}{}$= F(16) –F(1)

thus one of the possible value of k = 16