# How do you integrate #int x^3sqrt(1+x^2)# using integration by parts?

Rearranging the integral:

So, we see that integration by parts won't be necessary at all!

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To integrate ( \int x^3 \sqrt{1+x^2} ) using integration by parts, we choose ( u = x^3 ) and ( dv = \sqrt{1+x^2} , dx ).

Then, ( du = 3x^2 , dx ) and ( v = \int \sqrt{1+x^2} , dx ).

To find ( v ), make the substitution ( x = \sinh(t) ), then ( dx = \cosh(t) , dt ), and the integral becomes ( \int \cosh^2(t) , dt ).

This integral can be solved using the identity ( \cosh^2(t) = \frac{1 + \cosh(2t)}{2} ).

After integrating and substituting back ( x ), we can then apply the integration by parts formula:

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

Substitute ( u ), ( v ), ( du ), and ( dv ) into the integration by parts formula and solve for the integral.

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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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