The image contains a section on integration by parts, examples of integration formulas, and exercises with solutions. Here's a summary and analysis of the content:

Integration by Parts Formula:

If $f(x)$ and $g(x)$ are antiderivatives of $f'(x)$ and $g'(x)$, respectively, the formula is: $\int f(x) g'(x) dx = f(x) g(x) - \int f'(x) g(x) dx$

Immediate Applications: The table includes simplified formulas for $p(x)$ as a polynomial:

1. $\int p(x) e^x dx$
2. $\int p(x) \ln x dx$
3. $\int p(x) x^n dx$
4. $\int p(x) e^{-x} dx$

Exercises with Solutions:

1. $\int x e^x dx$:

Solution: $F(x) = e^x (x - 1) + c, \, c \in \mathbb{R}$

2. $\int e^{3x} dx$:

Solution: $F(x) = \frac{1}{3} e^{3x} + c, \, c \in \mathbb{R}$

3. $\int x \ln x dx$:

Solution: $F(x) = \frac{x^2}{2} (\ln x - \frac{1}{2}) + c, \, c \in \mathbb{R}$

4. $\int \ln x dx$:

Solution: $F(x) = x \ln x - x + c, \, c \in \mathbb{R}$

5. $\int x e^{-x} dx$:

Solution: $F(x) = -e^{-x} (x + 1) + c, \, c \in \mathbb{R}$

Would you like me to provide detailed derivations of these solutions, or do you have any specific questions about the integration techniques shown?

5 Related Questions:

1. How is the integration by parts formula derived?
2. Why is $u$-substitution essential for some integrals in these examples?
3. What are common scenarios where integration by parts is the best choice?
4. Can the formula $\int x e^x dx$ be solved in alternative ways?
5. What is the practical significance of these integrals in physics or engineering?

Tip:

When using integration by parts, always choose $u$ (from $\int u dv$) to simplify the integral as much as possible, often preferring $u$ as a logarithmic or polynomial function.