To solve the integral \(\int 10 \sin^2(x) \cos^3(x) \, dx\), we can use a substitution method.

Let's start by expressing the integral in a more convenient form:

\[
\int 10 \sin^2(x) \cos^3(x) \, dx
\]

We can use the substitution \( u = \sin(x) \), which means \( du = \cos(x) \, dx \). But first, let's rewrite the integral to make the substitution easier. Notice that \(\cos^2(x) = 1 - \sin^2(x)\), so we can rewrite \(\cos^3(x)\) as:

\[
\cos^3(x) = \cos(x) \cdot \cos^2(x) = \cos(x) \cdot (1 - \sin^2(x))
\]

Substituting this back into the integral:

\[
\int 10 \sin^2(x) \cos(x) (1 - \sin^2(x)) \, dx
\]

Now, use the substitution \( u = \sin(x) \), so \( du = \cos(x) \, dx \). The integral becomes:

\[
\int 10 u^2 (1 - u^2) \, du
\]

Expanding the integrand:

\[
\int 10 (u^2 - u^4) \, du
\]

Now, integrate each term:

\[
10 \int u^2 \, du - 10 \int u^4 \, du
\]

The integrals of \(u^2\) and \(u^4\) are:

\[
\int u^2 \, du = \frac{u^3}{3}
\]
\[
\int u^4 \, du = \frac{u^5}{5}
\]

So the integral becomes:

\[
10 \left(\frac{u^3}{3} - \frac{u^5}{5}\right) + C
\]

Finally, substitute back \( u = \sin(x) \):

\[
\frac{10}{3} \sin^3(x) - \frac{10}{5} \sin^5(x) + C
\]

Simplifying the coefficients:

\[
\frac{10}{3} \sin^3(x) - 2 \sin^5(x) + C
\]

Thus, the solution to the integral is:

\[
\int 10 \sin^2(x) \cos^3(x) \, dx = \frac{10}{3} \sin^3(x) - 2 \sin^5(x) + C
\]

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How would you approach the integral \(\int \sin^3(x) \cos^2(x) \, dx\)?
2. What is the integral of \(\int \sin^n(x) \cos^m(x) \, dx\) for general values of \(n\) and \(m\)?
3. How does the substitution method work in general for trigonometric integrals?
4. What are some other common trigonometric identities used in integration?
5. How do you integrate functions involving higher powers of sine and cosine?

**Tip:** When dealing with trigonometric integrals, look for opportunities to use identities or substitutions to simplify the integrand before integrating.

Learn how to solve the integral of 10 sin^2(x) cos^3(x) using the substitution method. This problem involves trigonometric integrals and demonstrates the application of trigonometric identities and substitution rules in integration. Suitable for advanced high school students and those studying calculus.

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