∫sin²(x)dx
(1/2)x-(1/2)sin(x)cos(x)+C
To integrate ∫sin²(x)dx, we can use the trigonometric identity:
sin²(x) = (1/2)(1-cos(2x))
Therefore, we can write:
∫sin²(x)dx = ∫(1/2)(1-cos(2x))dx
Using the linearity of integration, we can split it up as:
(1/2)∫dx – (1/2)∫cos(2x)dx
The first term integrates to:
(1/2)∫dx = (1/2)x + C
The second term involves a trigonometric function, which we can integrate as follows:
∫cos(2x)dx = (1/2)sin(2x) + C
Putting everything together, we get:
∫sin²(x)dx = (1/2)x – (1/4)sin(2x) + C
where C is the constant of integration. Therefore, this is the final answer.
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