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How do you integrate #(sinx)^2 dx#?

Answer 1

Ezra Adams

Try this:

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Answer 2

Christopher Agresta

To integrate (sinx)^2 dx, you can use the identity:

∫(sinx)^2 dx = ∫(1 - cos(2x))/2 dx

This identity arises from the double-angle identity for sine, which states that sin^2(x) = (1 - cos(2x))/2.

Using this identity, the integral simplifies to:

∫(1 - cos(2x))/2 dx

Which can then be further simplified to:

(1/2)∫(1 - cos(2x)) dx

Now, you can integrate each term separately:

∫(1 - cos(2x)) dx = ∫1 dx - ∫cos(2x) dx

Integrating, you get:

∫1 dx = x + C

And for ∫cos(2x) dx, you use the substitution method, where u = 2x, du = 2 dx:

∫cos(2x) dx = (1/2)∫cos(u) du

Now integrate cos(u):

(1/2)sin(u) + C

Replace u with 2x:

(1/2)sin(2x) + C

Putting it all together, the final result is:

(1/2)x - (1/4)sin(2x) + C

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.