Simplify and Integrate: Integrating sin²(x) Using Trigonometric Identity

∫sin²(x)dx

To integrate the function ∫sin²(x)dx, we can use a trigonometric identity to simplify it

To integrate the function ∫sin²(x)dx, we can use a trigonometric identity to simplify it. The trigonometric identity we can use is:

sin²(x) = (1 – cos(2x)) / 2

Let’s substitute this identity into the integral:

∫sin²(x)dx = ∫(1 – cos(2x)) / 2 dx

Now we can split this integral into two separate integrals:

∫(1/2 – cos(2x)/2) dx

The first integral, ∫(1/2)dx, is straightforward:

(1/2)∫dx = (1/2)x + C1

For the second integral, ∫(cos(2x)/2) dx, we can use the substitution method. Let’s substitute u = 2x:

du/dx = 2
dx = du/2

Now we can rewrite the integral in terms of u:

∫(cos(u)/2) (du/2) = (1/4)∫cos(u) du

Using the integral of the cosine function, we get:

(1/4)∫cos(u) du = (1/4)sin(u) + C2

Finally, substituting u back as 2x:

(1/4)sin(2x) + C2

Combining the two integrals, we get the final answer:

∫sin²(x)dx = (1/2)x – (1/4)sin(2x) + C

So, the integral of sin²(x)dx is (1/2)x – (1/4)sin(2x) + C, where C is the constant of integration.

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