What is the integral of $\int sin (x) cos (x)$ $int sin(x)cos(x)$ from 0 to 2?

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Answer 1 · 2016-01-31T13:36:03

$\int_{0}^{2} sin (x) cos (x) d x = \frac{{sin}^{2} (4)}{4}$ $int_0^2 sin(x)cos(x) dx = sin^2(4)/4$ .

Note that $sin (x) cos (x) = \frac{1}{2} sin (2 x)$ $sin(x)cos(x) = 1/2 sin(2x)$ .

Using that, we evaluate the above integral as:

$\int_{0}^{2} sin (x) cos (x) d x = \int_{0}^{2} \frac{1}{2} sin (2 x) d x$ $int_0^2 sin(x)cos(x) dx = int_0^2 1/2 sin(2x) dx$

$= {[\frac{{sin}^{2} (2 x)}{4}]}_{0}^{2}$ $= [sin^2(2x)/4]_0^2$

$= \frac{{sin}^{2} (4)}{4} - \frac{{sin}^{2} (0)}{4}$ $= sin^2(4)/4 - sin^2(0)/4$

$= \frac{{sin}^{2} (4)}{4}$ $= sin^2(4)/4$ .

What is the integral of ∫sin(x)cos(x)int sin(x)cos(x) from 0 to 2?