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

Then:

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

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

Now, integrate ( \frac{u}{2} \cos(u) ) with respect to ( u ) using integration by parts. Let ( dv = \cos(u) , du ) and ( u = \frac{u}{2} ). Then, ( v = \sin(u) ) and ( du = du ).

[ \int \frac{u}{2} \cos(u) , du = \frac{1}{2} \left( u \sin(u) - \int \sin(u) , du \right) ]

[ = \frac{1}{2} \left( u \sin(u) + \cos(u) \right) + C ]

Substitute back ( u = x^2 ):

[ = \frac{1}{2} \left( x^2 \sin(x^2) + \cos(x^2) \right) + C ]

Therefore, the integral of ( x^3 \cos(x^2) , dx ) is ( \frac{1}{2} \left( x^2 \sin(x^2) + \cos(x^2) \right) + 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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