Question

Find the value of integral in the branch b?

Answer 1

`= 1/2( e sin 1 - e cos 1 + 1)`

Explanation:

`I = int_1^e sin ( ln y ) dy`

Let:

• `ln y = x, qquad y = e^x, qquad 1/y dy = dx`

`I = int_0^1 sin ( x ) e^x dx`

Take the first result:

• `int e^x sin x \ dx = 1/2 e^x ( sin x - cos x) + C`

`I = ( 1/2 e^x ( sin x - cos x) )_0^1`

`= ( 1/2 e ( sin 1 - cos 1) ) - ( 1/2 e^0 ( sin 0 - cos 0) )`

`= 1/2( e sin 1 - e cos 1 + 1)`