Question

\(\int_0^2 \frac{3x+1}{x^2+4} dx\)

• A. \(\log(2\sqrt{2}) + \pi/4\)
• B. \(\log(2\sqrt{2}) + \pi/6\)
• C. \(\log(2\sqrt{2}) + \pi/8\)
• D. \(\log(2\sqrt{2}) + \pi/12\)

Hint:

Whenever you see \(\frac{ax+b}{x^2+c}\), split it into: \[ \frac{ax}{x^2+c}+\frac{b}{x^2+c} \] because each part integrates in a standard way.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Split into log and arctan parts.
Step 2: Key Formula or Approach:
Part 1: \(\int \frac{3x}{x^2+4}\). Part 2: \(\int \frac{1}{x^2+4}\).
Step 3: Detailed Explanation:
\(1.5\log(8/4) + 0.5\tan^{-1}(1) = \log(2^{1.5}) + \pi/8 = \log(2\sqrt{2}) + \pi/8\).
Step 4: Final Answer:
Matches option (C).