How do you find the integral of sin²x?

To find the integral of sin²x, we can use a useful trigonometric identity. The integral can be simplified using the identity:

Trigonometric Identity

We know that:

• sin²x = (1 - cos(2x)) / 2

Setting Up the Integral

Using this identity, we can rewrite the integral:

∫ sin²x dx = ∫ (1 - cos(2x)) / 2 dx

Breaking It Down

This can be separated into two simpler integrals:

• ∫ (1/2) dx - ∫ (1/2) cos(2x) dx

Calculating Each Integral

Now, we can integrate each part:

• ∫ (1/2) dx = (1/2)x
• ∫ (1/2) cos(2x) dx = (1/4) sin(2x)

Combining Results

Putting it all together, we have:

∫ sin²x dx = (1/2)x - (1/4) sin(2x) + C

Final Answer

Thus, the integral of sin²x is:

(1/2)x - (1/4) sin(2x) + C, where C is the constant of integration.