https://file.aiquickdraw.com/mathbot/file/17254712029069w61mrsq.jpg

The integral provided in the image is:

\[
\int 7 \sin^3(x) \cos(x) \, dx
\]

### Solution:

1. **Simplify the Integral**:
   Notice that the integral can be simplified by a substitution. Let’s use the substitution:
   \[
   u = \sin(x)
   \]
   Then,
   \[
   du = \cos(x) \, dx
   \]
   
2. **Rewrite the Integral**:
   Substitute \( u \) and \( du \) into the integral:
   \[
   \int 7 \sin^3(x) \cos(x) \, dx = 7 \int u^3 \, du
   \]

3. **Integrate**:
   The integral of \( u^3 \) with respect to \( u \) is:
   \[
   \frac{u^4}{4}
   \]
   Therefore:
   \[
   7 \int u^3 \, du = 7 \cdot \frac{u^4}{4} = \frac{7u^4}{4}
   \]

4. **Substitute Back**:
   Now substitute \( u = \sin(x) \) back into the expression:
   \[
   \frac{7u^4}{4} = \frac{7 \sin^4(x)}{4}
   \]

5. **Add the Constant of Integration**:
   Don’t forget to add the constant of integration \( C \):
   \[
   \frac{7 \sin^4(x)}{4} + C
   \]

### Final Answer:

\[
\int 7 \sin^3(x) \cos(x) \, dx = \frac{7 \sin^4(x)}{4} + C
\]

Would you like further details or have any questions?

Here are some related questions to consider:

1. What substitution would you use if the integral was \(\int 5 \sin^2(x) \cos(x) \, dx\)?
2. How would the solution change if the integral was \(\int \sin^3(x) \cos(x) \, dx\) without the constant 7?
3. What is the general formula for integrating \(\sin^n(x) \cos(x) \, dx\)?
4. How would you handle the integral \(\int \sin^3(x) \cos^2(x) \, dx\)?
5. What other integration techniques could apply if the powers of sine and cosine were different?

**Tip**: When dealing with powers of trigonometric functions, substitution using basic trigonometric identities often simplifies the integral significantly.

Evaluate the indefinite integral ∫ 7sin^3(x)cos(x) dx.

Explore the step-by-step solution for evaluating the indefinite integral ∫ 7sin^3(x)cos(x) dx. This problem involves calculus concepts like trigonometric substitution and integration techniques, suitable for students in Grades 11-12.

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