\int \cos ^3\left(x\right)\sin \left(x\right)dx

To solve the integral \(\int \cos^3(x) \sin(x) \, dx\), we can use a substitution method. Let's proceed step by step.

1. Let \(u = \cos(x)\). Then, the derivative of \(u\) with respect to \(x\) is:
   \[
   \frac{du}{dx} = -\sin(x)
   \]
   This implies:
   \[
   du = -\sin(x) \, dx
   \]
   or equivalently:
   \[
   -du = \sin(x) \, dx
   \]

2. Substitute \(u = \cos(x)\) and \(\sin(x) \, dx = -du\) into the integral:
   \[
   \int \cos^3(x) \sin(x) \, dx = \int u^3 (-du)
   \]
   Simplify the integral:
   \[
   \int \cos^3(x) \sin(x) \, dx = -\int u^3 \, du
   \]

3. Now, integrate \(u^3\):
   \[
   \int u^3 \, du = \frac{u^4}{4}
   \]

4. Don't forget to include the negative sign from the substitution:
   \[
   -\int u^3 \, du = -\frac{u^4}{4}
   \]

5. Substitute back \(u = \cos(x)\):
   \[
   -\frac{u^4}{4} = -\frac{(\cos(x))^4}{4} = -\frac{\cos^4(x)}{4}
   \]

6. Add the constant of integration \(C\):
   \[
   \int \cos^3(x) \sin(x) \, dx = -\frac{\cos^4(x)}{4} + C
   \]

So, the solution to the integral is:
\[
\boxed{-\frac{\cos^4(x)}{4} + C}
\]

Would you like further details or have any questions? Here are some related questions for further practice:

1. \(\int \cos^2(x) \sin(x) \, dx\)
2. \(\int \cos^3(x) \, dx\)
3. \(\int \sin^3(x) \cos(x) \, dx\)
4. \(\int \cos^5(x) \sin(x) \, dx\)
5. \(\int \sin^4(x) \cos(x) \, dx\)
6. \(\int \cos^2(x) \, dx\)
7. \(\int \cos(x) \, dx\)
8. \(\int \sin(x) \, dx\)

**Tip:** When integrating products of trigonometric functions, substitution often simplifies the problem significantly. Look for expressions where substitution can turn the integral into a simpler form.

Learn how to integrate cos^3(x) sin(x) with a detailed step-by-step solution using trigonometric substitution. This problem involves advanced integral calculus concepts and is suitable for advanced high school and college students.

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