# How do you find the integral of #intxcos(5x)dx#?

Use integration by parts.

Substituting these values into the integration by parts equation:

The remaining integral on the right is trivial:

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To find the integral of ( \int x \cos(5x) , dx ), you can use integration by parts. Apply the integration by parts formula:

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

Let: [ u = x ] [ dv = \cos(5x) , dx ]

Then, differentiate ( u ) to get ( du ), and integrate ( dv ) to get ( v ).

[ du = dx ] [ v = \frac{1}{5} \sin(5x) ]

Now, apply the integration by parts formula:

[ \int x \cos(5x) , dx = x \cdot \frac{1}{5} \sin(5x) - \int \frac{1}{5} \sin(5x) , dx ]

This integral can be straightforwardly integrated to get the final result.

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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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