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The image contains a list of calculus problems asking to compute indefinite integrals. I will now solve several of these for you.

Let's start with a few selected integrals:

### 7. \(\int \left( \sqrt{2y} - \frac{1}{\sqrt{2y}} \right) dy\)
Break the integral into two parts:
\[
\int \sqrt{2y} \, dy - \int \frac{1}{\sqrt{2y}} \, dy
\]

- For \(\int \sqrt{2y} \, dy\), let \(u = 2y\), so \(du = 2 dy\):
\[
\int \sqrt{2y} \, dy = \frac{1}{3} (2y)^{3/2}
\]

- For \(\int \frac{1}{\sqrt{2y}} \, dy\), it simplifies to:
\[
\int (2y)^{-1/2} \, dy = \frac{1}{\sqrt{2y}}
\]

Thus, the integral is:
\[
\frac{(2y)^{3/2}}{3} - \frac{1}{\sqrt{2y}} + C
\]

### 8. \(\int \frac{2dt}{3t^2 + 3}\)
Factor out the 3:
\[
\int \frac{2dt}{3(t^2 + 1)} = \frac{2}{3} \int \frac{dt}{t^2 + 1}
\]

We recognize this as a standard integral:
\[
\int \frac{dt}{t^2 + 1} = \tan^{-1}(t)
\]
So the result is:
\[
\frac{2}{3} \tan^{-1}(t) + C
\]

### 9. \(\int \sqrt{x} \, dx\)
For this, use the power rule. Rewrite the integrand:
\[
\int x^{1/2} \, dx = \frac{2}{3} x^{3/2} + C
\]

### 12. \(\int \frac{t^2 + 1}{t^3 + 3t} \, dt\)
Factor the denominator:
\[
\int \frac{t^2 + 1}{t(t^2 + 3)} \, dt
\]
Use substitution \(u = t^2 + 3\), \(du = 2t \, dt\), leading to:
\[
\int \frac{1}{t} \left(\frac{t^2 + 1}{t^2 + 3}\right) dt
\]
This requires partial fractions, and solving it yields the answer:
\[
\frac{1}{2} \ln |t^2 + 3| + C
\]

### 13. \(\int \frac{1}{\sqrt{9x^2 + 1}} \, dx\)
Use the substitution \(x = \frac{1}{3} \tan \theta\), then:
\[
\int \frac{1}{\sqrt{9x^2 + 1}} \, dx = \frac{1}{3} \sinh^{-1}(3x) + C
\]

These are detailed solutions for a few of the integrals. Let me know if you need more detailed steps or solutions to other specific problems.

### Follow-up questions:
1. How do we handle integrals involving trigonometric identities?
2. What is the method of substitution in integrals, and when do we use it?
3. Can you explain the process of using partial fractions in integrals?
4. How do we solve integrals involving inverse trigonometric functions?
5. What is the power rule for integration, and how does it apply?

### Tip:
Always simplify the integrand as much as possible before applying integration techniques. This makes solving easier and more manageable.

The user submitted an image containing a list of calculus problems asking to compute indefinite integrals.

Detailed solutions to a variety of indefinite integral problems, covering key concepts such as substitution, trigonometric identities, and the power rule. This resource is ideal for students studying calculus at the high school or undergraduate level.

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