Indefinite Integration
Integration by Parts
Grade 12
Question:
<p>Evaluate: \(\int x \sin x \sec^3 x dx\)</p>
<p>(A) \(\frac{1}{2}[\sec^2 x - \tan x] + C\)</p>
<p>(B) \(\frac{1}{2}[x \sec^2 x - \tan x] + C\)</p>
<p>(C) \(\frac{1}{2}[x \sec^2 x + \tan x] + C\)</p>
<p>(D) \(\frac{1}{2}[\sec^2 x + \tan x] + C\)</p>
Step-by-Step Solution
Key Concept: Recognize $\sin x \sec^3 x$ as $\tan x \sec^2 x$ and apply integration by parts.
<p><strong>Step 1:</strong> Rewrite $\sin x \sec^3 x = \sin x \sec^3 x = \tan x \sec^2 x$</p><p><strong>Step 2:</strong> Use integration by parts: $u = x$, $dv = \tan x \sec^2 x dx$</p><p><strong>Step 3:</strong> $du = dx$, $v = \frac{1}{2}\sec^2 x$</p><p><strong>Step 4:</strong> Apply integration by parts formula: $\int x \tan x \sec^2 x dx = \frac{x}{2}\sec^2 x - \frac{1}{2}\int \sec^2 x dx$</p><p><strong>Step 5:</strong> Evaluate: $\frac{x}{2}\sec^2 x - \frac{1}{2}\tan x + C = \frac{1}{2}[x \sec^2 x - \tan x] + C$</p><p>∴ Answer is B.</p>
Correct Answer: B