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Evaluate the integral ∫(5sin³(x) * x) dx. (Use c for the constant of integration.)

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Final answer:

To evaluate the given integral, we can use the integration by parts method by setting u = x and dv = 5sin³(x) dx. Simplifying the steps using the formula for integration by parts, the integral becomes -x*cos(x) + sin(x) + C, where C is the constant of integration.

Step-by-step explanation:

To evaluate the integral ∫(5sin³(x) * x) dx, we can use the integration by parts method. Let u = x and dv = 5sin³(x) dx. Taking the derivatives and antiderivatives, we have du = dx and v = -cos(x). Using the formula for integration by parts, ∫udv = uv - ∫vdu, we can evaluate the integral:

∫(5sin³(x) * x) dx = -x*cos(x) - ∫(-cos(x) dx)

Simplifying further, the integral becomes -x*cos(x) + sin(x) + C, where C is the constant of integration.

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