Question

$\int e^x\left(\frac{2+\sin 2x}{1+\cos 2x}\right)dx=$

• A. $e^x\cot x+c$
• B. $2e^x{\sec }^{2}x+c$
• C. $e^x\cos 2x+c$
• D. $e^x\tan x+c$

Answer 1

The correct option is D $e^x\tan x+c$

$\int e^x\left(\frac{2+\sin 2x}{1+\cos 2x}\right)dx$ $=\int e^x\left(\frac{2+2\sin x\cos x}{2{\cos }^{2}x}\right)dx$

$=\int e^x(\tan x+{\sec }^{2}x)dx=\int e^x(\tan x+(\tan x{)}^{\prime })dx=e^x\tan x+c$