# How do you integrate #int x^nsinx^(n)dx# using integration by parts?

answer

integration by parts

suppose:

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To integrate ( \int x^n \sin^n(x) , dx ) using integration by parts, we choose ( u = x^n ) and ( dv = \sin^n(x) , dx ).

Then, we differentiate ( u ) with respect to ( x ) to get ( du = n x^{n-1} , dx ), and we integrate ( dv ) to get ( v = -\frac{1}{n+1} \cos(x) \sin^{n-1}(x) ).

Now, applying the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

We get:

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

Simplifying, we have:

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

This integral can be further simplified or evaluated using other integration techniques depending on the specific values of ( n ).

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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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