Definite Integration
Integration by Parts
Grade 12
Question:
<p>The value of the integral \(\int_0^2 \frac{\log(x^2+2)}{(x^2+2)} dx\) is:</p>
<p>(a) \(\frac{2}{3}\tan^{-1}\sqrt{2} + \frac{5}{12}\log 2 - \frac{1}{4}\log 3\)</p>
<p>(b) \(\frac{2}{3}\tan^{-1}\sqrt{2} - \frac{5}{12}\log 2 - \frac{1}{12}\log 3\)</p>
<p>(c) \(\frac{2}{3}\tan^{-1}\sqrt{2} + \frac{5}{12}\log 2 + \frac{1}{12}\log 3\)</p>
<p>(d) \(\frac{2}{3}\tan^{-1}\sqrt{2} - \frac{5}{12}\log 2 + \frac{1}{12}\log 3\)</p>
Step-by-Step Solution
Key Concept: Apply integration by parts strategically, choosing logarithm as the first function to reduce complexity in the remaining integral.
<p>Use integration by parts with $u = \log(x^2+2)$ and $dv = \frac{dx}{x^2+2}$. Then $du = \frac{2x}{x^2+2}dx$ and $v = \frac{1}{\sqrt{2}}\tan^{-1}\left(\frac{x}{\sqrt{2}}\right)$. Evaluate the boundary term and the remaining integral, which requires partial fractions decomposition.</p>
Correct Answer: D