মান নির্ণয় কর:

  $\int \frac{e^{m \tan^{- 1}{x}}}{\left ( 1 + x^{2} \right )^{2}} dx$  

ধরি, $\tan ^{-1} \mathrm{x}=\theta \quad \therefore \mathrm{x}=\tan \theta \quad \therefore \mathrm{dx}=\sec ^{2} \theta \mathrm{d} \theta$

$\begin{array}{l} \therefore \mathrm{I}=\int \frac{\mathrm{e}^{\mathrm{m} \theta}}{\left(1+\tan ^{2} \theta\right)^{2}} \sec ^{2} \theta \mathrm{d} \theta \int \frac{\mathrm{e}^{\mathrm{m} \theta}}{\sec ^{2} \theta} \mathrm{d} \theta=\int \mathrm{e}^{\mathrm{m} \theta} \cos ^{2} \theta \mathrm{d} \theta=\frac{1}{2} \int \mathrm{e}^{\mathrm{m} \theta}(1+\cos 2 \theta) \mathrm{d} \theta \\ =\frac{1}{2} \int \mathrm{e}^{\mathrm{m} \theta} \mathrm{d} \theta+\frac{1}{2} \int \mathrm{e}^{\mathrm{m} \theta} \cos 2 \theta \mathrm{d} \theta\left[\because \int \mathrm{e}^{\mathrm{ax}} \cos \mathrm{bx} \mathrm{dx}=\frac{\mathrm{e}^{\mathrm{ax}}}{\mathrm{a}^{2}+\mathrm{b}^{2}}(\mathrm{a} \cos \mathrm{bx}+\mathrm{b} \sin \mathrm{bx})\right] \\ =\frac{1}{2 \mathrm{~m}} \mathrm{e}^{\mathrm{m} \theta}+\frac{1}{2} \cdot \frac{\mathrm{e}^{\mathrm{m} \theta}}{\mathrm{m}^{2}+2^{2}}\{\mathrm{~m} \cos 2 \theta+2 \sin \mathrm{2} \theta\}+\mathrm{c} \\ =\frac{1}{2 \mathrm{~m}} \mathrm{e}^{\mathrm{m} \tan ^{-1} \mathrm{x}}+\frac{\mathrm{e}^{\mathrm{m} \tan ^{-1} \mathrm{x}}}{\mathrm{m}^{2}+4}\left\{\mathrm{~m} \cos \left(2 \tan ^{-1} \mathrm{x}\right)+2 \sin \left(\mathrm{mtan}^{-1} \mathrm{x}\right)\right\}+\mathrm{c} \end{array}$