Definite Integral
General
Grade 12
Question:
If $\frac{d}{dx} f(x) = \frac{e^{\sin x}}{x}, x > 0$ & $\int_{1}^{4} \frac{2e^{\sin x^2}}{x} \, dx = f(k) - f(1)$, then find possible values of $k$.
Step-by-Step Solution
Key Concept: General
$x^2 = t \implies 2x dx = dt$. $\int_{1}^{4} \frac{2x \cdot e^{\sin(x^2)}}{x^2} \, dx = \int_{1}^{16} 2 \cdot \frac{e^{\sin t}}{t} \cdot \frac{dt}{2} = \int_{1}^{16} \frac{d}{dt} f(t) dt = f(16) - f(1)$. Thus $k = 16$.
Correct Answer: 16