Question

Evaluate the following integrals:
sin^2(x)dx from 0 to 2π,
|sin x| dx from 0 to 2π,
(x^n)(e^−x)dx from 0 to ∞
sin(ax)sin(bx)dx from 0 to π
Consider all possible cases (n is a positive integer)

Answer 1

1) hint : sin^2 x = (1-cos2x)/2

Answer 2

2) divide 2 case for :
Area1 = int (sinx) dx [0,pi]
area2 = int(-sinx)dx [pi,2pi]
the total area is .........

Answer 3

Why is it negative sinx?

Answer 4

because the absolut function must be positive number, so the area of under x-axes should give a (-) sign too, so that (-)(-) be postive

Answer 5

That makes sense, thanks

Answer 6

let's look this graph