To find the integral of $\cos^2(x) \sin(x) \, dx$, we can use a substitution method.

Step-by-Step Solution

1. Substitute: Let $u = \cos(x)$.

   Then, the derivative $du = -\sin(x) \, dx$, or equivalently, $-du = \sin(x) \, dx$.

2. Rewrite the Integral: Using the substitution $u = \cos(x)$, we can rewrite $\cos^2(x) \sin(x) \, dx$ in terms of $u$ and $du$:

   $\cos^2(x) \sin(x) \, dx = u^2 (-du) = -u^2 \, du$

3. Integrate: Now, integrate with respect to $u$:

   $\int -u^2 \, du = -\frac{u^3}{3} + C$

4. Substitute Back: Substitute $u = \cos(x)$ back into the expression:

   $-\frac{u^3}{3} + C = -\frac{\cos^3(x)}{3} + C$

Final Answer

$\int \cos^2(x) \sin(x) \, dx = -\frac{\cos^3(x)}{3} + C$

This is the integral of $\cos^2(x) \sin(x)$ with respect to $x$.

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Would you like more details on this method or have any other questions?

Related Questions

1. How would you integrate $\sin^2(x) \cos(x) \, dx$?
2. What is the integral of $\sin^3(x) \, dx$?
3. How can integration by parts be applied to trigonometric integrals?
4. What are the common substitutions used for integrals involving trigonometric functions?
5. How can the reduction formula be applied to integrals like $\int \sin^n(x) \cos^m(x) \, dx$?

Tip: For integrals involving powers of trigonometric functions, look for opportunities to use substitutions that simplify the expression.