# What is the integral of #int sin^5 (x) dx#?

The answer is

We need

The integral is

Perform the substitution

Therefore,

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To find the integral of ( \int \sin^5(x) , dx ), you can use the reduction formula for powers of sine functions:

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

Using this formula repeatedly, you can reduce the power of ( \sin(x) ) until it reaches a known integral. However, this process can become quite complex and iterative for higher powers of ( \sin(x) ).

Another approach is to use trigonometric identities to rewrite ( \sin^5(x) ) in terms of lower powers of ( \sin(x) ) and ( \cos(x) ), and then integrate each term separately. This method can be more straightforward than using the reduction formula directly.

Alternatively, you can also use numerical integration methods if an exact solution is not necessary or feasible.

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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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