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Question: Answered & Verified by Expert
$\int_0^3 \frac{3 x+1}{x^2+9} d x$ is equal to :
MathematicsDefinite IntegrationAP EAMCETAP EAMCET 2003
Options:
  • A $\log (2 \sqrt{2})+\frac{\pi}{12}$
  • B $\log (2 \sqrt{2})+\frac{\pi}{2}$
  • C $\log (2 \sqrt{2})+\frac{\pi}{6}$
  • D $\log \left(2(\sqrt{2})+\frac{\pi}{3}\right.$
Solution:
2011 Upvotes Verified Answer
The correct answer is: $\log (2 \sqrt{2})+\frac{\pi}{12}$


$=\frac{3}{2}\left[\log \left(x^2+9\right)\right]_0^3+\frac{1}{3}\left[\tan ^{-1} \frac{x}{3}\right]_0^3$
$=\frac{3}{2}[\log 18-\log 9]+\frac{1}{3}\left[\tan ^{-1}(1)-\tan ^{-1}(0)\right]$
$=\frac{3}{2}[\log 2]+\frac{\pi}{12}$
$=\log (2 \sqrt{2})+\frac{\pi}{12}$

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