Definite Integration Question 67
Question 67
- Evaluate
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Solution:
- Let
Put
Put
$$ \begin{aligned} \Rightarrow & I=2 \pi \int_{1 / \sqrt{3}}^{\sqrt{3}} \frac{2 d u}{1+3 u^{2}}=\frac{4 \pi}{3}\left[\sqrt{3} \tan ^{-1} \sqrt{3} u\right]{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \ & =\frac{4 \pi}{\sqrt{3}}\left(\tan ^{-1} 3-\tan ^{-1} 1\right)=\frac{4 \pi}{\sqrt{3}} \tan ^{-1} \frac{1}{2} \ \therefore & \int{-\pi / 3}^{\pi / 3} \frac{\pi+4 x^{3}}{2-\cos |x|+\frac{\pi}{3}} d x=\frac{4 \pi}{\sqrt{3}} \tan ^{-1} \frac{1}{2} \end{aligned} $$
