# How do you integrate #(x^3cos(x^2)dx)#?

I found:

I used Substitution and Integration by Parts:

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To integrate ( x^3 \cos(x^2) , dx ), you can use a substitution method. Let ( u = x^2 ), then ( du = 2x , dx ). Solving for ( dx ), we have ( dx = \frac{du}{2x} ). Substituting these into the integral, we get:

[ \int x^3 \cos(x^2) , dx = \int \frac{u}{2} \cos(u) , du ]

This can be simplified further by using integration by parts. Let ( dv = \cos(u) , du ), then ( v = \sin(u) ). Applying the integration by parts formula:

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

We get:

[ \frac{u \sin(u)}{2} - \int \frac{\sin(u)}{2} , du ]

Integrating ( \frac{\sin(u)}{2} ), we get:

[ \frac{u \sin(u)}{2} + \frac{\cos(u)}{2} + C ]

Finally, replacing ( u ) back with ( x^2 ), we have:

[ \frac{x^2 \sin(x^2)}{2} + \frac{\cos(x^2)}{2} + 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.

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