# How do you integrate #int (x^2(x^3 + 1)^3)dx#?

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To integrate ( \int x^2(x^3 + 1)^3 , dx), you can use the substitution method. Let ( u = x^3 + 1 ). Then, ( du = 3x^2 , dx ). Rearranging, ( \frac{1}{3} , du = x^2 , dx ).

Now, substitute ( u ) and ( \frac{1}{3} , du ) into the integral:

[ \int x^2(x^3 + 1)^3 , dx = \int \frac{1}{3}u^3 , du ]

This simplifies to:

[ \frac{1}{3} \int u^3 , du ]

Then integrate ( u^3 ) with respect to ( u ):

[ \frac{1}{3} \cdot \frac{1}{4} u^4 + C ]

Substitute back for ( u ):

[ \frac{1}{12}(x^3 + 1)^4 + C ]

Where ( C ) is the constant of integration.

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