how do you integrate pi (integral) sin squared x ?

Question

aka.... pi S (sin(x))^2

the "S" is the integral sin lol.

I dunno why but I'm stuck on this.

Answer 1

(sin(x))^2 = 1- (cos(x)^2)

Because sin(x + pi/2) = sinx*cos(pi/2) + cosx*sin(pi/2) = cos x

and pi is the cycle of cos^2 and sin^2

The integral from zero to pi of (sin x)^2 and that of (cos x)^2 are equal.

Now, 2*int(o to pi) (sin x)^2 dx =
= int(o to pi) (sin x)^2 dx +
+ int(o to pi) (cos x)^2 dx = int(o to pi) 1 dx = pi

Divide LHS and RHS by 2 and get
int(o to pi) (sin x)^2 dx = pi/2

Answer 2

This is a reasonable problem to be stuck on--it's tricky to do unless you try the right approach.

The right approach is to use a trigonometric identity; namely, (sin (x))^2 = (1 - cos 2x) / 2. With this, the integral becomes

pi S [(1 - cos 2x) / 2] dx

= (pi / 2)S (1 - cos 2x) dx

= (pi / 2)(x - (1/2) sin 2x) + C

= (pi / 2)x - (pi / 4) sin 2x + C