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You've uploaded an image containing several integral problems. Let's go through some of these integrals step-by-step.

### Problem 1: \( \int \frac{dx}{x^2} \)
This is a straightforward integral. We can solve it by using the power rule for integration:

\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\]

Here, \( n = -2 \), so:

\[
\int x^{-2} \, dx = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C
\]

### Problem 2: \( \int \frac{dx}{x^2 \sqrt{x}} \)
First, simplify the expression. Since \( \sqrt{x} = x^{1/2} \), we can rewrite the integrand as:

\[
\frac{1}{x^2 \cdot x^{1/2}} = \frac{1}{x^{5/2}}
\]

Now, apply the power rule:

\[
\int x^{-5/2} \, dx = \frac{x^{-3/2}}{-3/2} = -\frac{2}{3x^{3/2}} + C
\]

### Problem 3: \( \int \frac{dh}{\sqrt{2gh}} \) (where \( g \) is a constant)
This is an integral of the form:

\[
\int h^{-1/2} \, dh
\]

Apply the power rule for integration:

\[
\int h^{-1/2} \, dh = \frac{h^{1/2}}{1/2} = 2\sqrt{h}
\]

Thus, the integral becomes:

\[
\frac{2}{\sqrt{2g}} \cdot \sqrt{h} + C = \frac{\sqrt{2h}}{g} + C
\]

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Would you like me to continue solving the remaining integrals or explain any of the steps in more detail?

Here are five related questions that expand on these concepts:

1. How do you handle integrals involving trigonometric functions, such as \( \int \sin 2x \, dx \)?
2. How can the substitution method be used in solving integrals?
3. What are common techniques to integrate rational functions, like those with \( x^2 + 1 \)?
4. How can partial fractions be applied to integrals involving polynomials?
5. What role do special functions (like logarithms and exponentials) play in integration?

**Tip:** Always simplify the integrand before integrating—it often leads to a more straightforward solution!

求下列不定积分 (List of integrals from the image)

Solve indefinite integrals of various forms, including power and rational functions. This page explores key calculus concepts such as the power rule, substitution method, and integrals involving square roots. Ideal for students at an undergraduate level.

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