How do you find the integral of # sin^3[x]dx#?
Regarding the initial integral:
With substitution, the second integral is as follows:
When we combine everything, we obtain the following outcome:
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To find the integral of ( \sin^3(x) ) with respect to ( x ), you can use the reduction formula for powers of sine. [ \int \sin^n(x) , dx = -\frac{1}{n} \sin^{n-1}(x) \cos(x) + \frac{n-1}{n} \int \sin^{n-2}(x) , dx ] Applying this formula with ( n = 3 ), we get [ \int \sin^3(x) , dx = -\frac{1}{3} \sin^2(x) \cos(x) + \frac{2}{3} \int \sin(x) , dx ] Integrating ( \sin(x) ) gives ( -\cos(x) ), so the final result is [ \int \sin^3(x) , dx = -\frac{1}{3} \sin^2(x) \cos(x) - \frac{2}{3} \cos(x) + C ] where ( C ) is the constant of integration.
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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.
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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