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Here is the problem: integral of x squared e to the power x cubed).

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Find the indefinite integral (antiderivative) of x squared e to the power x cubed) with respect to x.

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Start with the integral.

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Substitute u equals (x cubed), so du equals (3x squared) dx.

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Exponential rule: the integral of eᵘ du equals eᵘ (divide by the inner slope).

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Substitute back u equals (x cubed).

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Simplify and add the constant of integration.

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So the answer is e to the power x cubed/3 plus C.

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An antiderivative F satisfies F prime of x equals f(x). Because the derivative of any constant is 0, every antiderivative differs by a constant C.

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Differentiated the answer and compared it with the integrand at 8 sample points, which checks out.