Indefinite Integration
Integration by Recognition
Grade 12
Question:
<p>\(\int \frac{1+x\cos x}{x(1+x^2 e^{2\sin x})} dx\) is equal to</p>
<p>(A) \(\log|xe^{\sin x}| + \frac{1}{2}\log|1 - x^2 e^{2\sin x}| + C\)</p>
<p>(B) \(\log|xe^{\sin x}| - \frac{1}{2}\log|1 - x^2 e^{2\sin x}| + C\)</p>
<p>(C) \(\log|xe^{\sin x}| - \frac{1}{2}\log|1 + x^2 e^{2\sin x}| + C\)</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Recognize the derivative of $xe^{\sin x}$ in the numerator and split the fraction strategically.
<p>Rewrite the integrand by noting the derivative of $xe^{\sin x}$ is $e^{\sin x}(1 + x\cos x)$. The integral splits into two parts: $\int \frac{e^{\sin x}(1+x\cos x)}{xe^{\sin x}} dx - \int \frac{x^2 e^{2\sin x} \cos x}{x(1+x^2 e^{2\sin x})} dx$, which evaluates to the given answer.</p>
Correct Answer: C