Indefinite Integration
Exponential and Trigonometric Functions
Grade 12
Question:
<p><span>\(\int (g(x) + 1) \sin x\)</span> dx is equal to</p>
<p>(A) <span>\(\frac{e^x}{2}(\sin x - \cos x) + C\)</span></p>
<p>(B) <span>\(\frac{e^x}{5}(2\sin x - \cos x) + C\)</span></p>
<p>(C) <span>\(\frac{e^x}{5}(\sin 2x - \cos 2x) + C\)</span></p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Recognize the exponential-trigonometric integral pattern and apply integration by parts or standard formulas.
<p>Express <span>$g(x) + 1$</span> from the matrix exponential and integrate <span>$(g(x) + 1)\sin x$</span> using integration by parts or the formula for integrals of exponential times trigonometric functions.</p>
Correct Answer: B