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Let's break down and solve these integrals step by step using substitution methods.

### Q1) \(\int \frac{\sec(\ln(x)) \tan(\ln(x))}{x} \, dx\)

Here, a substitution of \( u = \ln(x) \) will simplify the integral.

- \( du = \frac{1}{x} dx \), so \( dx = x \, du \).

The integral becomes:
\[
\int \sec(u) \tan(u) \, du
\]
The derivative of \(\sec(u)\) is \(\sec(u) \tan(u)\), so the integral is:
\[
\sec(u) + C = \sec(\ln(x)) + C
\]
Thus, the solution to Q1 is:
\[
\boxed{\sec(\ln(x)) + C}
\]

---

### Q2) \(\int \sqrt{e^x} \, e^x \, dx\)

We simplify the expression inside the integral:
\[
\sqrt{e^x} = e^{x/2}
\]
Thus, the integral becomes:
\[
\int e^{x/2} \cdot e^x \, dx = \int e^{3x/2} \, dx
\]

Now, use substitution: let \( u = \frac{3x}{2} \), so \( du = \frac{3}{2} dx \), or \( dx = \frac{2}{3} du \).

The integral becomes:
\[
\frac{2}{3} \int e^u \, du = \frac{2}{3} e^u + C
\]
Substitute back \( u = \frac{3x}{2} \):
\[
\boxed{\frac{2}{3} e^{3x/2} + C}
\]

---

### Q3) \(\int \frac{e^{\sqrt{x}}}{2\sqrt{x}} \, dx\)

Here, use the substitution \( u = \sqrt{x} \), so that \( du = \frac{1}{2\sqrt{x}} dx \), or equivalently \( dx = 2\sqrt{x} \, du \).

The integral simplifies to:
\[
\int e^u \, du = e^u + C
\]
Substitute back \( u = \sqrt{x} \):
\[
\boxed{e^{\sqrt{x}} + C}
\]

---

### Q4) \(\int \sqrt{4x + 3} \, dx\)

Let \( u = 4x + 3 \), then \( du = 4 \, dx \), or \( dx = \frac{du}{4} \).

The integral becomes:
\[
\frac{1}{4} \int \sqrt{u} \, du = \frac{1}{4} \int u^{1/2} \, du
\]
Integrating \( u^{1/2} \):
\[
\frac{1}{4} \cdot \frac{2}{3} u^{3/2} + C = \frac{1}{6} (4x + 3)^{3/2} + C
\]
Thus, the solution is:
\[
\boxed{\frac{1}{6} (4x + 3)^{3/2} + C}
\]

---

Would you like further clarification on any of these integrals?

### Follow-up questions:
1. What are the key steps in applying substitution for each problem?
2. How does changing the variable simplify each integral?
3. Why does \(\sec(u)\) differentiate to \(\sec(u)\tan(u)\)?
4. Could these problems be solved with other methods like integration by parts?
5. What are other common integrals involving \(\sec\) and \(\tan\)?

#### Tip:
When performing substitution, always remember to adjust both the differential and the bounds (if definite integrals) accordingly.

Integrals requiring substitution techniques, including trigonometric and exponential functions.

This page provides solutions to calculus integration problems using substitution techniques. Learn how to solve integrals involving sec(ln(x)) tan(ln(x)), exponential functions like e^x, and other expressions. Suitable for undergraduate calculus students.

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