Let's solve each of the given integration problems one by one:

a) $\int \frac{\ln x}{x} \, dx$

To solve this, we can use the substitution $u = \ln x$, then $du = \frac{1}{x} \, dx$.

$\int \frac{\ln x}{x} \, dx = \int u \, du = \frac{u^2}{2} + C = \frac{(\ln x)^2}{2} + C$

So, the solution is:

$\boxed{\frac{(\ln x)^2}{2} + C}$

b) $\int e^x \sin(e^x) \, dx$

Let $u = e^x$, then $du = e^x \, dx$. This transforms the integral into:

$\int e^x \sin(e^x) \, dx = \int \sin(u) \, du$

The integral of $\sin(u)$ is $-\cos(u)$, so:

$\int \sin(u) \, du = -\cos(u) + C = -\cos(e^x) + C$

Thus, the solution is:

$\boxed{-\cos(e^x) + C}$

c) $\int x^3 \sqrt{x^2 + 11} \, dx$

Let $u = x^2 + 11$. Then, $du = 2x \, dx$, or $\frac{du}{2} = x \, dx$.

We rewrite $x^3 \sqrt{x^2 + 11}$ as $x^2 \cdot x \sqrt{x^2 + 11}$. Then, $x^2 = u - 11$:

$\int x^3 \sqrt{x^2 + 11} \, dx = \int (u - 11) \sqrt{u} \cdot \frac{du}{2}$

This simplifies to:

$\frac{1}{2} \int (u - 11) u^{1/2} \, du = \frac{1}{2} \int (u^{3/2} - 11u^{1/2}) \, du$

Integrate each term separately:

$\frac{1}{2} \left( \frac{2}{5} u^{5/2} - 11 \cdot \frac{2}{3} u^{3/2} \right) + C$

Simplifying this:

$\frac{1}{5} u^{5/2} - \frac{11}{3} u^{3/2} + C$

Substitute $u = x^2 + 11$:

$\boxed{\frac{1}{5} (x^2 + 11)^{5/2} - \frac{11}{3} (x^2 + 11)^{3/2} + C}$

d) $\int \frac{e^{\ln x}}{\cos^2 x} \, dx$

First, notice that $e^{\ln x} = x$:

$\int \frac{x}{\cos^2 x} \, dx = \int x \sec^2 x \, dx$

We can use integration by parts for this integral. Let $u = x$ and $dv = \sec^2 x \, dx$. Then, $du = dx$ and $v = \tan x$:

$\int x \sec^2 x \, dx = x \tan x - \int \tan x \, dx$

The integral of $\tan x$ is $-\ln |\cos x|$:

$x \tan x + \ln |\cos x| + C$

Thus, the solution is:

$\boxed{x \tan x + \ln |\cos x| + C}$

If you have any questions or need further details on any of these integrations, feel free to ask! Here are some related questions to explore the topic further:

1. What is the general method for integrating functions of the form $\frac{\ln x}{x}$?
2. How do you identify appropriate substitutions for integrals involving composite functions?
3. What are the common techniques used in integration by parts?
4. Can you derive the formula for integrating $e^x \sin(e^x)$ without substitution?
5. How do you handle integrals involving square roots of quadratic expressions?
6. What are some examples of integrals that require a combination of substitution and parts?
7. How does the integration of trigonometric functions differ from polynomial functions?
8. What are some common mistakes to avoid when performing integration by parts?

Tip: Always verify your integration results by differentiating your answer to see if you obtain the original integrand. This ensures the accuracy of your solution.