_0^ /2 x x ,dx ^2 x + 3 x + 2 = - Vidyadip

MCQ

$\int_0^{\pi /2} {\frac{{\sin x\cos x\,dx}}{{{{\cos }^2}x + 3\cos x + 2}}} = $

A
$\log \left( {\frac{8}{9}} \right)$
✓
$\log \left( {\frac{9}{8}} \right)$
C
$\log (8 \times 9)$
D
None of these

✓

Answer

Correct option: B.

$\log \left( {\frac{9}{8}} \right)$

b
(b) Let $I = \int_0^{\pi /2} {\frac{{\sin x\cos x.dx}}{{{{\cos }^2}x + 3\cos x + 2}}} $

We put $\cos x = t \Rightarrow - \sin x\,dx = dt,$ then

$I = \int_0^1 {\frac{{t.dt}}{{{t^2} + 3t + 2}} = \int_0^1 {\left[ {\frac{2}{{t + 2}} - \frac{1}{{t + 1}}} \right]} } \,dt$

$ = [2\log (t + 2) - \log (\,t + 1)]_0^1$

$ = [2\log 3 - \log 2 - 2\log 2]$

$ = [2\log 3 - 3\log 2] = [\log 9 - \log 8] $

$= \log \left( {\frac{9}{8}} \right)$.

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