Let $\frac{d}{dx}f(x) = \frac{e^{\sin x}}{x}$, $x > 0$. If $\int_1^{2e^{\sin x^2}} \frac{2e^{\sin x^2}}{x}dx = F(k) - F(1)$, then find one of the possible value of $k/4$.
Step-by-Step Solution
Key Concept: Use substitution to transform a difficult integral into a standard form, then apply the fundamental theorem of calculus.
The integral $\int_1^4 \frac{2e^{\sin t^2}}{t} dx$ is evaluated using substitution. Let $u = \sin t^2$, then $du = 2t\cos t^2 \, dt$. Rewriting the integrand and evaluating from $t=1$ to $t=4$, we get $[F(t)]_1^{16} = F(16) - F(1) = \frac{k}{4}$. Solving yields $k = 16$.
Correct Answer: 4