Indefinite Integration
Integration by substitution with exponential functions
Grade 12
Question:
<p>Evaluate \(\int \frac{e^x(1+x)}{\cos^2(e^x x)} dx\)</p>
<p>(a) \(-\cot(e^x) + C\)</p>
<p>(b) \(\tan(xe^x) + C\)</p>
<p>(c) \(\tan(e^x) + C\)</p>
<p>(d) \(\cot(e^x) + C\)</p>
Step-by-Step Solution
Key Concept: Use substitution with exponential-trigonometric composition to simplify the integral using standard integral formulas.
<p><strong>Solution:</strong> Let $u = e^x x$, then $du = e^x(1+x)dx$.</p><p>The integral becomes $\int \frac{du}{\cos^2 u} = \int \sec^2 u \, du = \tan u + C = \tan(e^x) + C$</p>
Correct Answer: C