# Evaluate the integral by making appropriate substitution. \int \cos^3 x \sin x dx

## Question:

Evaluate the integral by making appropriate substitution.

{eq}\int \cos^3 x \sin x dx {/eq}

## Integration using Substitution:

Integration can be made simpler by applying prior substitution to the integrand. We must take note, however, that the differential variable also changes. Upon successful integration over the transformed integrand, we must express the final answer in terms of the original variables.

## Answer and Explanation:

Evaluate the integral by applying the substitution, {eq}\displaystyle u = \cos x {/eq}, wherein the differential variable becomes {eq}\displaystyle du = -\sin xdx {/eq}. We apply the substitution and then proceed with the integration.

{eq}\begin{align} \displaystyle \int\cos^3 x\sin xdx &= -\int u^3du\\ &= - \frac{u^4}{4} +C\\ \text{Revert the substitution.}\\ &= - \frac{\cos ^4 x}{4}+C \end{align} {/eq}

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