Question

The value $\int e^{\tan \theta }(\sec \theta -\sin \theta )d\theta$ is equal to?

• A. $-e^{\tan \theta }\sin \theta +C$
• B. $e^{\tan \theta }\sin \theta +C$
• C. $e^{\tan \theta }\sec \theta +C$
• D. $e^{\tan \theta }\cos \theta +C$

Answer 1

Answer: (A)

$-e^{\tan \theta }\sin \theta +C$

$\int e^{\tan \theta }(\sec \theta -\sin \theta )d\theta$

$\int e^{\tan \theta }(-\sec \theta +{\sec }^2\theta \cos \theta d\theta )$

$=\int e^{\tan \theta }(-\sin \theta +\cos \theta {\sec }^2\theta d\theta )$

$\int e^{f\left(x\right)}(-g(x)+f^{\prime }(x\left)g^{\prime }\right(x\left)dx\right)$

$=-e^{f\left(x\right)}g\left(x\right)$

$=e^{\tan \theta }\sin \theta$