Evaluate ∫ from 0 to π of sin^2 x dx Step by Step

Step by step

The question

Question: find the integral from 0 to π of sin² x.

∫ from 0 to π of sin² x dx =

1 Use the half-angle rule

The half-angle rule: sin squared of x is 1 minus cos 2x, all over 2.

It changes a square into a plain cos.

sin² x = 1 − cos 2x2

Use it in the integral.

= ∫ from 0 to π of 1 − cos 2x2 dx

Split the fraction: 1/2, minus 1/2 cos 2x.

= ∫ from 0 to π of (12 − 12 cos 2x) dx

Careful!

The power rule does not work on sin squared: the derivative of sin x is not there.

Use the half-angle rule.

Careful: not ∫ sin² x dx = sin³ x3 ✗

2 Integrate

cos 2x integrates to sin 2x/2: divide by the 2 inside.

∫ cos 2x dx = sin 2x2

No + C: top minus bottom.

= [ 12x − sin 2x4 ] from 0 to π

3 Top minus bottom

sin of a whole number of halves of a turn is 0.

sin 0 = sin π = sin 2π = 0

Put in π, then take away the same with 0.

The sin parts are 0.

= 12 × π − 12 × 0

So the integral is π over 2, about 1.57.

= π2 ≈ 1.57

4 Check

Let's check with a calculator: it adds up 1000 thin strips.

That gives about 1.57, the same as our answer.

Right!

1000 thin strips: ≈ 1.57

Let's look back.

First: use the half-angle rule.

Then: integrate.

Then: top minus bottom.

Last: check.

Answer