The image contains several indefinite and definite integrals that need to be evaluated. Here is the approach to solve each of them.

Indefinite Integrals (Problems 1-10)

1. $\int x^4 \, dx$
   Use the power rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$.

2. $\int \frac{1}{x^2} \, dx$
   Rewrite as $\int x^{-2} \, dx$ and apply the power rule.

3. $\int 5 \sqrt{x^3} \, dx$
   Rewrite $\sqrt{x^3}$ as $x^{3/2}$ and apply the power rule.

4. $\int \frac{\tan x}{\sec x} \, dx$
   Simplify $\frac{\tan x}{\sec x}$ and integrate.

5. $\int \frac{x^2 + 3}{\sqrt{x}} \, dx$
   Separate into two integrals: $\int \frac{x^2}{\sqrt{x}} \, dx + \int \frac{3}{\sqrt{x}} \, dx$, then simplify each term.

6. $\int \frac{(1 - 2x)^2}{\sqrt{x}} \, dx$
   Expand $(1 - 2x)^2$ and then divide each term by $\sqrt{x}$ before integrating term by term.

7. $\int \frac{1}{\sqrt{x}(\sqrt{x} - 1)^2} \, dx$
   Consider a substitution to simplify the expression, such as $u = \sqrt{x} - 1$.

8. $\int \left( x \sqrt{x} - \frac{1}{\sqrt{x}} \right) \, dx$
   Separate into two integrals and simplify each term using the power rule.

9. $\int (x \sqrt{x} - 5)^2 \, dx$
   Expand $(x \sqrt{x} - 5)^2$ and then integrate term by term.

10. $\int \left( x^{1/3} + x^{2/3} \right)^2 \, dx$
    Expand $\left( x^{1/3} + x^{2/3} \right)^2$ and then integrate term by term.

Definite Integrals

1. $\int_{4}^{9} \frac{2}{\sqrt{x}} \, dx$
   Integrate $\frac{2}{\sqrt{x}} = 2x^{-1/2}$ and apply the limits from 4 to 9.

2. $\int_{2}^{4} (-x^2 + 6x - 8) \, dx$
   Integrate each term separately and then evaluate from 2 to 4.

3. $\int_{0}^{2} \sin x \, d(\sin x)$
   Recognize $d(\sin x)$ as $\cos x \, dx$, then rewrite and integrate.

4. $\int_{0}^{\pi/2} 2 \cos \theta \, d\theta$
   Integrate $2 \cos \theta$ and evaluate from 0 to $\pi/2$.

5. $\int_{x}^{x^2} (v^2 + \sin 3v) \, dv$
   Treat as a function of $x$, where the limits are variable. Apply the Fundamental Theorem of Calculus for functions with variable limits.

Would you like the detailed solutions to each of these integrals?

Related Questions

1. How do you apply substitution for integrals with square roots?
2. What is the Fundamental Theorem of Calculus for variable limits?
3. How do you approach integrals involving trigonometric identities?
4. What is the power rule for integration, and when can it be used?
5. How do you solve integrals with polynomial expressions?

Tip

When faced with complex integrals, breaking down the expression term by term or considering substitutions often simplifies the process.