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Definite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration2 Marks
MCQ
The value of $\int_0^\pi \frac{d x}{5+4 \cos x}$ is
  • A
    $2 \pi$
  • B
    $\frac{3 \pi}{2}$
  • C
    $\frac{5 \pi}{4}$
  • ✓
    $\frac{\pi}{3}$
✓

Answer

Correct option: D.
$\frac{\pi}{3}$
(D)
Put $\tan \frac{x}{2}= t$
$\therefore \quad d x=\frac{2 dt }{1+ t ^2}$ and $\cos x=\frac{1- t ^2}{1+ t ^2}$
$\therefore \quad \int_0^\pi \frac{ d x}{5+4 \cos x}=\int_0^{\infty} \frac{2 dt }{9+ t ^2}$
$\begin{array}{l}=\left[\frac{2}{3} \tan ^{-1}\left(\frac{ t }{3}\right)\right]_0^{\infty} \\ =\frac{2}{3}\left(\tan ^{-1} \infty-0\right) \\ =\frac{2}{3} \cdot \frac{\pi}{2}=\frac{\pi}{3}\end{array}$

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