Let's solve each of these integrals step by step with detailed explanations.

1. $\int \frac{dx}{x^2}$

Rewrite the integrand: $\int x^{-2} \, dx$

Use the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1$

Here, $n = -2$: $\int x^{-2} \, dx = \frac{x^{-2+1}}{-2+1} + C = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C$

So, $\int \frac{dx}{x^2} = -\frac{1}{x} + C$

2. $\int x^{\sqrt{2}} \, dx$

Use the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1$

Here, $n = \sqrt{2}$: $\int x^{\sqrt{2}} \, dx = \frac{x^{\sqrt{2}+1}}{\sqrt{2}+1} + C$

So, $\int x^{\sqrt{2}} \, dx = \frac{x^{\sqrt{2}+1}}{\sqrt{2}+1} + C$

3. $\int \frac{1}{x^\pi} \, dx$

Rewrite the integrand: $\int x^{-\pi} \, dx$

Use the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1$

Here, $n = -\pi$: $\int x^{-\pi} \, dx = \frac{x^{-\pi+1}}{-\pi+1} + C$

So, $\int \frac{1}{x^\pi} \, dx = \frac{x^{1-\pi}}{1-\pi} + C$

4. $\int x\sqrt{x} \, dx$

Rewrite $\sqrt{x}$ as $x^{1/2}$: $\int x \cdot x^{1/2} \, dx = \int x^{1 + 1/2} \, dx = \int x^{3/2} \, dx$

Use the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1$

Here, $n = \frac{3}{2}$: $\int x^{3/2} \, dx = \frac{x^{3/2 + 1}}{3/2 + 1} + C = \frac{x^{5/2}}{5/2} + C = \frac{2}{5} x^{5/2} + C$

So, $\int x\sqrt{x} \, dx = \frac{2}{5} x^{5/2} + C$

If you have any questions or need further details, feel free to ask.

Here are 5 related questions you can ask:

1. How do you solve $\int \frac{1}{x} \, dx$?
2. Can you explain the substitution method in integration?
3. What is the integration by parts formula?
4. How do you integrate trigonometric functions?
5. Can you solve a definite integral, like $\int_0^1 x^2 \, dx$?

Tip: Always remember to add the constant of integration $C$ when solving indefinite integrals.