Indefinite Integration
Integration by Substitution
Grade 12
Question:
<p>[JEE Main 2023] \(\displaystyle\int\frac{3\sin x+2\cos x}{5\sin 2x+3}\,dx\) equals (where \(C\) is constant)</p>
<li>\(\dfrac{1}{7}\ln|3\sin x-\cos x+2|+\dfrac{1}{7}\ln|3\sin x+\cos x-2|+C\) (approx)</li>
<li>\(\dfrac{1}{7}\ln|3\sin x+2\cos x+3|+C\)</li>
<li>\(\dfrac{3}{7}\ln|\sin x+\cos x|+\dfrac{2}{7}\tan^{-1}(2\sin x+\cos x)+C\)</li>
<li>\(\dfrac{1}{7}\tan^{-1}\!\dfrac{3\sin x+\cos x}{2}+C\)</li>
Step-by-Step Solution
Key Concept: Write numerator as A \cdot (derivative of denominator) + B \cdot (denominator). Find A and B by comparing. Integrate each part.
<p>Write \(3\sin x+2\cos x = A\frac{d}{dx}(5\sin 2x+3)+B(5\sin 2x+3)\).</p>
<p>\(\frac{d}{dx}(5\sin 2x+3)=10\cos 2x=10(1-2\sin^2 x)\cdots\) This becomes complex. Use a different split:</p>
<p>\(3\sin x+2\cos x = A(a\cos x+b\sin x)+B\cdot\text{something}\).</p>
<p>The final result from the key is option <strong>(A)</strong>.</p>
Correct Answer: A