# How do you find the antiderivative of #int (sin^7x)dx# from #[-1,1]#?

We use the following Result to find the reqd. anti-derivative :

The Proof of the Result is easy, so let us not bother about it.

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To find the antiderivative of ( \int \sin^7(x) , dx ) from ([-1, 1]), follow these steps:

- Use the power-reducing identity for sine to rewrite ( \sin^7(x) ) in terms of lower powers of sine.
- Integrate the resulting expression term by term.
- Evaluate the antiderivative at the upper and lower bounds, and subtract the value at the lower bound from the value at the upper bound to find the definite integral.

Applying the power-reducing identity for sine, ( \sin^2(x) = \frac{1 - \cos(2x)}{2} ), repeatedly, you can reduce ( \sin^7(x) ) to terms involving sine and cosine with powers less than 7. Then integrate term by term.

This process will give you the antiderivative. Finally, evaluate this antiderivative at the upper and lower bounds and subtract the result at the lower bound from the result at the upper bound to find the definite integral over the interval ([-1, 1]).

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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.

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.

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.

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