Evaluate: `inte^x\ ((2+sin2x)/(1+cos2x))\ dx`

Options

A. `e^(x)tanx+C`
B. `e^(x)+tanx+C`
C. `2e^(x)tanx+C`
D. `e^(x)tan2x+C`

Answer

A

Explanation

Let `l=inte^(x)((2+sin2x)/(1+cos2x))dx`
`implies l=inte^(x)((2)/(1+cos2x)+(sin2x)/(1+cos2x))dx`
`impliesl l=inte^(x)[(2)/(2cos^(2)x)+(2sinxcosx)/(2cos^(2)x)]dx`
`implies l=inte^(x)[sec^(2)x+tanx]dx`
`implies l=inte^(x)(tanx+sec^(2)x)dx`
`implies l =e^(x) tanx+C`
`[because inte^(x)[f(x)+f'(x)]dx=e^(x)f(x)=C]`