8. integrate 1/(1 - 2cos x) dx

10. integrate 1/(2 + 3cos x) dx

Let's solve each integral one by one.

### 1. Integral of \(\frac{1}{1 - 2\cos x} \, dx\)

The integral can be solved using the Weierstrass substitution:

Let \( t = \tan\left(\frac{x}{2}\right) \), which gives us:
\[
\cos x = \frac{1 - t^2}{1 + t^2}, \quad dx = \frac{2 \, dt}{1 + t^2}
\]

Substituting these into the integral:

\[
\int \frac{1}{1 - 2\cos x} \, dx = \int \frac{1}{1 - 2\left(\frac{1 - t^2}{1 + t^2}\right)} \cdot \frac{2 \, dt}{1 + t^2}
\]

Simplifying the integrand:

\[
= \int \frac{2}{(1 + t^2) - 2(1 - t^2)} \, dt = \int \frac{2}{2t^2} \, dt = \int \frac{1}{t^2} \, dt
\]

Integrating:

\[
\int \frac{1}{t^2} \, dt = -\frac{1}{t} + C = -\frac{1}{\tan\left(\frac{x}{2}\right)} + C
\]

So the integral is:

\[
\boxed{-\frac{1}{\tan\left(\frac{x}{2}\right)} + C}
\]

### 2. Integral of \(\frac{1}{2 + 3\cos x} \, dx\)

Similarly, we use the Weierstrass substitution here:

\[
\int \frac{1}{2 + 3\cos x} \, dx = \int \frac{2}{(2 + 3\left(\frac{1 - t^2}{1 + t^2}\right))(1 + t^2)} \, dt
\]

Simplifying the integrand:

\[
= \int \frac{2}{\frac{2(1 + t^2) + 3(1 - t^2)}{1 + t^2}} \, dt = \int \frac{2(1 + t^2)}{(5 + t^2)} \, dt
\]

Further simplification:

\[
= \int \left(\frac{2}{5 + t^2} + \frac{2t^2}{5 + t^2}\right) dt
\]

Now, we separate and integrate:

\[
\int \frac{2}{5 + t^2} dt + \int \frac{2t^2}{5 + t^2} dt
\]

The first term can be integrated directly:

\[
\int \frac{2}{5 + t^2} dt = \frac{2}{\sqrt{5}} \arctan\left(\frac{t}{\sqrt{5}}\right)
\]

For the second term:

\[
\int \frac{2t^2}{5 + t^2} dt = \int 2 \, dt - \int \frac{10}{5 + t^2} dt = 2t - \frac{2\sqrt{5}}{\sqrt{5}} \arctan\left(\frac{t}{\sqrt{5}}\right)
\]

Combining both:

\[
= \frac{2}{\sqrt{5}} \arctan\left(\frac{\tan\left(\frac{x}{2}\right)}{\sqrt{5}}\right) + C
\]

So, the integral is:

\[
\boxed{\frac{2}{\sqrt{5}} \arctan\left(\frac{\tan\left(\frac{x}{2}\right)}{\sqrt{5}}\right) + C}
\]

Would you like further details or have any questions?

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Here are five related questions to consider:
1. How do you use the Weierstrass substitution to simplify trigonometric integrals?
2. What are the alternative methods to evaluate integrals involving trigonometric functions?
3. How can you apply partial fraction decomposition to integrals with trigonometric functions?
4. What is the role of the arctangent function in evaluating certain integrals?
5. How can integration by parts be applied to trigonometric integrals?

**Tip:** Always check for simpler substitutions before using advanced techniques like Weierstrass substitution for integration.

Learn how to integrate 1/(1 - 2cos x) dx and 1/(2 + 3cos x) dx using Weierstrass substitution and detailed step-by-step solutions. These integrals involve trigonometric functions and require advanced techniques. Suitable for advanced high school students and college-level calculus.

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