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How do you find #intsin^2(x)dx#?

Answer 1

Luke Alvarez

I would use integration by parts and a little final trick...!

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Answer 2

Paisley Johnson

To find ( \int \sin^2(x) , dx ), you can use trigonometric identities. One way to solve it is by using the double-angle identity for sine: ( \sin^2(x) = \frac{1 - \cos(2x)}{2} ). Then, you integrate the resulting expression with respect to ( x ).

[ \int \sin^2(x) , dx = \int \frac{1 - \cos(2x)}{2} , dx ]

Now, split the integral into two separate integrals:

[ \frac{1}{2} \int 1 , dx - \frac{1}{2} \int \cos(2x) , dx ]

Integrating each term separately:

[ = \frac{1}{2} \int 1 , dx - \frac{1}{2} \cdot \frac{\sin(2x)}{2} + C ]

[ = \frac{1}{2}x - \frac{1}{4}\sin(2x) + C ]

Where ( C ) is the constant of integration.

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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.