Indefinite Integration
Standard Integration Formulas
Grade 12
Question:
<p>\(\int \frac{a+b\cos x}{(b+a\cos x)^2} dx\) is equal to</p>
<p>(A) \(\frac{\sin x}{b+a\cos x} + C\)</p>
<p>(B) \(\frac{\cos x}{b+a\sin x} + C\)</p>
<p>(C) \(\frac{\sin x}{b+a\sin x} + C\)</p>
<p>(D) \(\frac{\cos x}{b+a\cos x} + C\)</p>
Step-by-Step Solution
Key Concept: Recognize that the numerator is related to the derivative of the denominator and use the formula for reciprocal derivatives.
<p>The numerator $a + b\cos x$ can be expressed as a derivative of the denominator. Using the standard formula for integrals of the form $\int \frac{f'(x)}{[f(x)]^2} dx = -\frac{1}{f(x)}$, we get $\frac{\sin x}{b+a\cos x} + C$.</p>
Correct Answer: A