Question

The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.

Options

• `e^x tan  x/2 +  c`

• e^x sin x + c

• e^x cos x + c

• `e^x sin  x/2 + c`

Answer 1

The value of `int e^x((1 + sinx)/(1 + cosx))dx` is `underlinebb(e^x  tan  x/2 +  c)`.

Explanation:

I = `int e^x((1 + sinx)/(1 + cosx))dx`

= `int e^x[1/(1 + cosx) + sinx/(1 + cosx)]dx`

= `int e^x[1/(2cos^2(x/2)) + (2sin(x/2)cos(x/2))/(2cos^2(x/2))]dx`

= `int e^x[1/2sec^2(x/2) + tan(x/2)]dx`

Now `d/dx tan(x/2) = 1/2sec^2  x/2`

Using the formula:

`int e^x[f(x) + f^'(x)]dx`

= `e^x f(x) + c`

= `e^x tan(x/2) + c`