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Here is the problem: integrate sin(4x)x.

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Find the indefinite integral (antiderivative) of sin(4 x) x with respect to x.

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Start with the integral.

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Integration by parts (tabular method). Differentiate the polynomial x until it reaches 0, integrate sin(4 x) the same number of times, and combine with alternating signs.

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Combine the alternating products.

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Simplify and add the constant of integration.

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So the answer is -(x times cos(4x) / 4) plus sin(4x) / 16 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.