The uploaded image contains two integrals that need to be solved. Let's analyze and solve them one by one.

Problem (b)

The integral is: $\int \frac{x}{e^{3x}} \, dx$

Solution:

1. Rewrite the integrand: $\frac{x}{e^{3x}} = x e^{-3x}$

2. Integration by parts: Use the formula: $\int u \, dv = uv - \int v \, du$ Let:

   • $u = x$ (hence $du = dx$),
   • $dv = e^{-3x} dx$ (hence $v = \frac{e^{-3x}}{-3} = -\frac{1}{3} e^{-3x}$).

3. Substitute into the integration by parts formula: $\int x e^{-3x} dx = u v - \int v \, du$ $= \left( x \cdot -\frac{1}{3} e^{-3x} \right) - \int \left( -\frac{1}{3} e^{-3x} \right) dx$ $= -\frac{x}{3} e^{-3x} + \frac{1}{3} \int e^{-3x} dx$

4. Solve the remaining integral: $\int e^{-3x} dx = \frac{e^{-3x}}{-3} = -\frac{1}{3} e^{-3x}$

5. Combine the results: $\int x e^{-3x} dx = -\frac{x}{3} e^{-3x} - \frac{1}{9} e^{-3x} + C$

6. Final answer: $\boxed{\int \frac{x}{e^{3x}} dx = -\frac{x}{3} e^{-3x} - \frac{1}{9} e^{-3x} + C}$

Problem (c)

The integral is: $\int \sin^2 x \cos^3 x \, dx$

Solution:

1. Simplify the powers: Rewrite $\cos^3 x$ as $\cos^2 x \cdot \cos x$: $\int \sin^2 x \cos^3 x \, dx = \int \sin^2 x \cos^2 x \cos x \, dx$

2. Use the Pythagorean identity: Replace $\cos^2 x$ with $1 - \sin^2 x$: $\int \sin^2 x \cos^3 x \, dx = \int \sin^2 x (1 - \sin^2 x) \cos x \, dx$

3. Substitution: Let $u = \sin x$, so $du = \cos x \, dx$.

   Substituting: $\int \sin^2 x (1 - \sin^2 x) \cos x \, dx = \int u^2 (1 - u^2) \, du$

4. Expand the integrand: $\int u^2 (1 - u^2) \, du = \int (u^2 - u^4) \, du = \int u^2 \, du - \int u^4 \, du$

5. Integrate term by term: $\int u^2 \, du = \frac{u^3}{3}, \quad \int u^4 \, du = \frac{u^5}{5}$

   So: $\int u^2 (1 - u^2) \, du = \frac{u^3}{3} - \frac{u^5}{5} + C$

6. Back-substitute $u = \sin x$: $\int \sin^2 x \cos^3 x \, dx = \frac{\sin^3 x}{3} - \frac{\sin^5 x}{5} + C$

7. Final answer: $\boxed{\int \sin^2 x \cos^3 x \, dx = \frac{\sin^3 x}{3} - \frac{\sin^5 x}{5} + C}$

Would you like more details or clarifications? Here are 5 related questions:

1. How does integration by parts work in other scenarios?
2. What are common substitution methods for trigonometric integrals?
3. How does the Pythagorean identity simplify trigonometric integrals?
4. Can you solve integrals involving higher powers of $\sin x$ and $\cos x$?
5. What are other approaches to solving integrals of the form $xe^{ax}$?

Tip: Always look for substitutions or identities to simplify the integrand before starting complex integration techniques!