Let $f(x) = \int \sin^4 x (1 + x\cos x \cdot \ln x + \sin x)dx$ and $f\left(\frac{\pi}{2}\right) = \frac{\pi^2}{4}$, then the value of $|\cos f(\pi))|$ is
Step-by-Step Solution
Key Concept: The integrand $F(x) + xF'(x)$ can be written as $(xF(x))'$, allowing direct integration without computing $F'(x)$ explicitly.
Given $f(x) = \int x^{\sin x}(1 + x\cos x \ln x + \sin x)dx$ with $F(x) = x^{\sin x} = e^{\sin x \ln x}$, recognize that $f(x) = \int(F(x) + xF'(x))dx = xF(x) + C = x \cdot x^{\sin x} + C$. This uses the product rule in reverse, where the integrand contains both the function and its derivative in a special form.
Correct Answer: [A-s] [B-t] [C-r] [D-q]