_0^3 3 x+1 x^2+9 d x= - Vidyadip

MCQ

$\int_0^3 \frac{3 x+1}{x^2+9} d x=$

✓
$\log (2 \sqrt{2})+\frac{\pi}{12}$
B
$\log (2 \sqrt{2})+\frac{\pi}{2}$
C
$\log (2 \sqrt{2})+\frac{\pi}{6}$
D
$\log (2 \sqrt{2})+\frac{\pi}{3}$

✓

Answer

Correct option: A.

$\log (2 \sqrt{2})+\frac{\pi}{12}$

(A)
$\int_0^3 \frac{3 x+1}{x^2+9} d x=\frac{3}{2} \int_0^3 \frac{2 x}{x^2+9} d x+\int_0^3 \frac{d x}{x^2+9}$
$=\left[\frac{3}{2} \log \left(x^2+9\right)+\frac{1}{3} \tan ^{-1}\left(\frac{x}{3}\right)\right]_0^3$
$=\frac{3}{2}(\log 18-\log 9)+\frac{1}{3}\left(\frac{\pi}{4}\right)$
$=\frac{3}{2} \log 2+\frac{\pi}{12}=\log (2 \sqrt{2})+\frac{\pi}{12}$

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