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.