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Here is the problem: integrate cos(x) cubed.

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Find the indefinite integral (antiderivative) of cos(x) cubed with respect to x.

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Start with the integral.

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Odd power: write cos cubed u equals cos u times (1 minus sin squared u) to the power 1, then substitute u equals sin.

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Substitute u equals sin(x), so du equals (cos(x)) dx.

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Power rule: the integral of x to the power n dx equals x to the power n plus 1/(n plus 1).

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Substitute back u equals sin(x).

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Substitute u equals sin(x), so du equals (cos(x)) dx.

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Integrate term by term.

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Simplify and add the constant of integration.

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So the answer is sin(x) minus sin(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.