The problem focuses on solving the integral of $\sin^3(x)e^{\cos(x)} \cos(x) \, dx$ using substitution and integration by parts.

Key Steps:

1. Substitution: Let $u = \cos(x)$, so $du = -\sin(x) \, dx$. This transforms the integral into terms of $u$.

2. Rewriting the Integral: The original integral becomes: $\int \sin^3(x) e^{\cos(x)} \cos(x) \, dx = -\int (1 - u^2) e^u \, du$

3. Integration by Parts (First Time): Decompose the integral $\int (1 - u^2)e^u \, du$ using: $f(u) = (1 - u^2), \quad g'(u) = e^u, \quad g(u) = e^u.$

   Apply the integration by parts formula: $\int f(u)g'(u) \, du = f(u)g(u) - \int f'(u)g(u) \, du$ Compute $f'(u)$, then solve the resulting integrals step-by-step.

4. Second Integration by Parts: For the resulting integral, use $f(u) = 2u$ and $g'(u) = e^u$, and compute similarly.

5. Final Expression: The solution simplifies to: $\int \sin^3(x)e^{\cos(x)}\cos(x) \, dx = \left( \cos(x)^2 - 2\cos(x) + 1 \right)e^{\cos(x)} + C$

Would you like me to delve deeper into any step, or help with verification? Let me know!

Related Questions:

1. What is the significance of substitution in integral problems?
2. How does integration by parts work conceptually?
3. Why is $-\sin(x)dx$ critical when substituting $u = \cos(x)$?
4. Can we directly integrate $\sin^3(x)e^{\cos(x)} \cos(x)$? Why or why not?
5. How does the exponential function $e^u$ simplify integration by parts?

Tip: Always double-check your substitutions when solving complex integrals to avoid errors!