A Step-by-Step Guide to Finding the Integral of Sin²(x) with Respect to x

∫sin²(x)dx

To find the integral of sin²(x) with respect to x, we need to use a trigonometric identity

To find the integral of sin²(x) with respect to x, we need to use a trigonometric identity. Specifically, we can use the identity cos(2x) = 1 – 2sin²(x).

Starting with the given integral:
∫sin²(x)dx

We can rewrite sin²(x) as (1-cos(2x))/2 using the mentioned trigonometric identity. This gives us:
∫(1 – cos(2x))/2 dx

Now, let’s separate this integral into two parts:
∫(1/2 – cos(2x)/2) dx

We can split this into two separate integrals:
∫1/2 dx – ∫cos(2x)/2 dx

The integral of 1/2 dx is simply (1/2)x + C, where C is the constant of integration.

For the second integral, we need to apply the power rule. The integral of cos(2x) can be found by using a substitution. Let’s substitute u = 2x, which means du/dx = 2, or du = 2dx. This gives us:
(1/2)∫cos(u) du

The integral of cos(u) is sin(u), so substituting back the original variable x, we have:
(1/2)∫cos(2x) dx = (1/2)sin(2x) + C

Combining the two parts, we get the final result:
∫sin²(x)dx = (1/2)x – (1/4)sin(2x) + C

Thus, this is the detailed solution for the integral of sin²(x) with respect to x.

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