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  4. ∫ e tan θ( sec θ- sin θ) d θ is equal to

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Q. $\int e^{\tan \theta}(\sec \theta-\sin \theta) d \theta$ is equal to

Integrals

Solution:

$I=\int e^{\tan \theta}(\sec \theta-\sin \theta) d \theta $
$=\int e^{\tan \theta}\left(\frac{\sec \theta-\sin \theta}{\sec ^2 \theta}\right) \sec ^2 \theta d \theta$
$I=\int e^{\tan \theta}\left(\frac{1}{\sec \theta}-\frac{\sin \theta}{\sec ^2 \theta}\right) \sec ^2 \theta d \theta $
$=\int e^{\tan \theta}\left(\frac{1}{\sqrt{1+\tan ^2 \theta}}-\frac{\tan \theta}{\left(1+\tan ^2 \theta\right)^{3 / 2}}\right) \sec ^2 \theta d \theta$
$= e ^{\tan \theta} \cdot \frac{1}{\sqrt{1+\tan ^2 \theta}}= e ^{\tan \theta} \cdot \cos \theta+C$