Let's solve these integrals one by one.

1. $\int \frac{1}{1 + 2\sin x} \, dx$

We use the Weierstrass substitution $t = \tan\left(\frac{x}{2}\right)$, where $\sin x = \frac{2t}{1+t^2}$ and $dx = \frac{2 \, dt}{1 + t^2}$.

Substituting: $\int \frac{1}{1 + 2\sin x} \, dx = \int \frac{1}{1 + \frac{4t}{1+t^2}} \cdot \frac{2 \, dt}{1+t^2}$ Simplifying, and using partial fractions if necessary, this will yield an integral in terms of $t$, which can be integrated. The final result can be expressed back in terms of $x$.

2. $\int \frac{1}{\sin x + \cos x} \, dx$

Start by multiplying the numerator and denominator by $\frac{1}{\sqrt{2}}$, we have: $\int \frac{1}{\sin x + \cos x} \, dx = \int \frac{\sqrt{2}}{\sqrt{2}\sin x + \sqrt{2}\cos x} \, dx = \int \frac{\sqrt{2}}{\sqrt{2}\left(\frac{\sin x + \cos x}{\sqrt{2}}\right)} \, dx$ Let $u = \sin x + \cos x$, $du = (\cos x - \sin x) \, dx$. The integral simplifies, and we integrate with respect to $u$.

3. $\int \frac{\tanh(x)}{\cosh(x) + 64\text{sech}(x)} \, dx$

Use the identity $\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$ and express the integral in terms of $u = \cosh(x)$, where $du = \sinh(x) \, dx$. The integral becomes: $\int \frac{\sinh(x)}{\cosh(x) \left(\cosh(x) + 64 \cdot \frac{1}{\cosh(x)}\right)} \, dx$ Simplifying and integrating in terms of $u$.

4. $\int \frac{1}{4 + 3\cosh(x)} \, dx$

Let $u = \tanh\left(\frac{x}{2}\right)$, so $\cosh(x) = \frac{1 + u^2}{1 - u^2}$, and substitute. Simplify the integral and solve in terms of $u$.

5. $\int \frac{1}{2 + \sin x} \, dx$

Similarly to the first integral, use the Weierstrass substitution $t = \tan\left(\frac{x}{2}\right)$ and express the integral in terms of $t$, then solve and revert back to $x$.

6. $\int \frac{1}{3\sin x + 4\cos x} \, dx$

We can solve this by using the Weierstrass substitution or express $3\sin x + 4\cos x$ as $R\sin(x + \alpha)$ and then integrate.

7. $\int \frac{\sinh(x)}{4\tanh(x) - \cosh(x)\text{sech}(x)} \, dx$

Simplify using identities for hyperbolic functions, then substitute $u = \tanh(x)$, and integrate in terms of $u$.

8. $\int \frac{1}{4 + 3\sinh(x)} \, dx$

Use a similar substitution as with the $\cosh(x)$ integral, but this time with $u = \tanh\left(\frac{x}{2}\right)$.

Detailed Solution Example:

Let's solve one in more detail:

4. $\int \frac{1}{4 + 3\cosh(x)} \, dx$

Let $u = \tanh\left(\frac{x}{2}\right)$, then: $\cosh(x) = \frac{1 + u^2}{1 - u^2}, \quad dx = \frac{2 \, du}{1-u^2}$ The integral becomes: $\int \frac{1}{4 + 3 \cdot \frac{1+u^2}{1-u^2}} \cdot \frac{2 \, du}{1-u^2}$ Simplify the expression and integrate with respect to $u$. The result can be converted back to $x$.

Would you like further details or have any questions on these integrals? Here are five related questions:

1. How does Weierstrass substitution help in trigonometric integrals?
2. Can you explain the integration of hyperbolic functions?
3. What is the significance of partial fractions in integral calculus?
4. How can the method of substitution simplify complex integrals?
5. What are common trigonometric identities used in integration?

Tip: For integrals involving trigonometric functions, always consider simplifying using identities or substitutions that can convert the expression into a more manageable form.