∫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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