Indefinite Integration
Integration by Parts
Grade 12
Question:
<p>[JEE Advanced 2008] Using the standard result \(\displaystyle\int e^x\{f(x)+f'(x)\}\,dx = e^x f(x)+C\):</p>
<p>If \(\displaystyle\int e^x\frac{1+\sin x}{1+\cos x}\,dx = e^x g(x)+C\), then \(g(x)\) is</p>
<li>\(\sin x\)</li>
<li>\(\cos x\)</li>
<li>\(\tan\dfrac{x}{2}\)</li>
<li>\(\cot\dfrac{x}{2}\)</li>
Step-by-Step Solution
Key Concept: Write (1+sinx)/(1+cosx) = [½sec^2(x/2) + tan(x/2)]. Identify f(x)=tan(x/2) and f'(x)=½sec^2(x/2).
<p>Using half-angle: \(1+\cos x=2\cos^2(x/2)\), \(\sin x=2\sin(x/2)\cos(x/2)\):</p>
<p>\[\frac{1+\sin x}{1+\cos x}=\frac{1}{2\cos^2(x/2)}+\frac{\sin(x/2)}{\cos(x/2)}=\frac12\sec^2\frac x2+\tan\frac x2\]</p>
<p>Let \(f(x)=\tan(x/2)\Rightarrow f'(x)=\frac12\sec^2(x/2)\).</p>
<p>The integrand \(= e^x(f+f')\Rightarrow\int e^x\frac{1+\sin x}{1+\cos x}\,dx=e^x\tan\frac x2+C\).</p>
<p>So \(g(x)=\tan(x/2)\). Answer: <strong>(C)</strong></p>
Correct Answer: C