(a) Find: ∫ (x^2+1)/(( x^2+2)(2 x^2+1)) d x

$\displaystyle \int \frac{\mathrm{x}^{2}+1}{\left(\mathrm{x}^{2}+2\right)\left(2 \mathrm{x}^{2}+1\right)} d \mathrm{x}=\frac{1}{3} \int \frac{1}{\mathrm{x}^{2}+2} d \mathrm{x}+\frac{1}{3} \int \frac{1}{2 \mathrm{x}^{2}+1} d \mathrm{x} \quad$ (Using Partial Fractions)

\[=\frac{1}{3} \cdot \frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{\mathrm{x}}{\sqrt{2}}\right)+\frac{1}{3} \cdot \frac{1}{\sqrt{2}} \tan ^{-1}(\sqrt{2} \mathrm{x})+C \]

Let $\displaystyle \mathrm{I}=\int_{0}^{\pi} \frac{\mathrm{x} \boldsymbol{\operatorname { t a n }} \mathrm{x}}{\sec \mathrm{x}+\tan \mathrm{x}} \mathrm{dx}$

\[\Rightarrow \mathrm{I}=\int_{0}^{\pi} \frac{(\pi-\mathrm{x}) \tan \mathrm{x}}{\sec \mathrm{x}+\tan \mathrm{x}} d \mathrm{x} \]