Question

$\int e^{\sin x}.\left(\frac{\sin x+1}{\sec x}\right)dx$ is equal to

• A. $\sin x.e^{\sin x}+c$
• B. $\cos x.e^{\sin x}+c$
• C. $e^{\sin x}+c$
• D. $e^{\sin x}(\sin x+1)+c$

Answer 1

Answer: (A)

Let I = $\int e^{\sin x}\left[\frac{\sin x+1}{\sec x}\right]dx=\int e^{\sin x}\left[\sin x+1\right]\cos xdx$
Put $\sin x=t$
$\Rightarrow \cos xdx=dt$
$\therefore I=\int e^t\left[1+t\right]dt=e^{t}t+c=e^{\sin x}.\sin x+c$