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What's the integral of sin(x)*cos(x)?

The integral of sin(x)*cos(x) is (1/2)sin^2(x) + C.

To find the integral of sin(x)*cos(x), we can use the trigonometric identity sin(2x) = 2sin(x)cos(x). Rearranging this equation, we get sin(x)*cos(x) = (1/2)sin(2x).

Therefore, the integral of sin(x)*cos(x) can be written as the integral of (1/2)sin(2x) dx. Using the substitution u = 2x, we get du/dx = 2 and dx = (1/2)du. Substituting these into the integral, we get:

∫sin(x)*cos(x) dx = ∫(1/2)sin(2x) dx
= (1/2)∫sin(u) du
= -(1/2)cos(u) + C
= -(1/2)cos(2x) + C

However, we can also use the trigonometric identity cos^2(x) + sin^2(x) = 1 to rewrite the integral as:

∫sin(x)*cos(x) dx = (1/2)∫sin(2x) dx
= -(1/4)cos(2x) + C
= (1/2)sin^2(x) + C

Therefore, the integral of sin(x)*cos(x) can be written as either -(1/2)cos(2x) + C or (1/2)sin^2(x) + C.

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