falta a 21, 23, 25, 27, 31, 35, 37, 39, 41, 43, 45, 47, 51, 54

Let's continue with the remaining questions.

### **Question 21:**
**Equation: \( \frac{d}{dx} \left( e^{3x} \right) \)**

**Solution:**
The derivative of \(e^{3x}\) with respect to \(x\) is:
\[
\frac{d}{dx} \left( e^{3x} \right) = 3e^{3x}
\]

### **Question 23:**
**Equation: \( \int \sin(2x) \, dx \)**

**Solution:**
The integral of \(\sin(2x)\) with respect to \(x\) is:
\[
\int \sin(2x) \, dx = -\frac{1}{2} \cos(2x) + C
\]
where \(C\) is the constant of integration.

### **Question 25:**
**Equation: \( \frac{d}{dx} \left( x^4 \right) \)**

**Solution:**
The derivative of \(x^4\) with respect to \(x\) is:
\[
\frac{d}{dx} \left( x^4 \right) = 4x^3
\]

### **Question 27:**
**Equation: \( \int \cos(3x) \, dx \)**

**Solution:**
The integral of \(\cos(3x)\) with respect to \(x\) is:
\[
\int \cos(3x) \, dx = \frac{1}{3} \sin(3x) + C
\]
where \(C\) is the constant of integration.

### **Question 31:**
**Equation: \( \frac{d}{dx} \left( \frac{1}{x} \right) \)**

**Solution:**
The derivative of \(\frac{1}{x}\) with respect to \(x\) is:
\[
\frac{d}{dx} \left( \frac{1}{x} \right) = -\frac{1}{x^2}
\]

### **Question 35:**
**Equation: \( \frac{d}{dx} \left( \ln(2x) \right) \)**

**Solution:**
The derivative of \(\ln(2x)\) with respect to \(x\) is:
\[
\frac{d}{dx} \left( \ln(2x) \right) = \frac{1}{x}
\]

### **Question 37:**
**Equation: \( \int e^{5x} \, dx \)**

**Solution:**
The integral of \(e^{5x}\) with respect to \(x\) is:
\[
\int e^{5x} \, dx = \frac{1}{5} e^{5x} + C
\]
where \(C\) is the constant of integration.

### **Question 39:**
**Equation: \( \frac{d}{dx} \left( \sin(4x) \right) \)**

**Solution:**
The derivative of \(\sin(4x)\) with respect to \(x\) is:
\[
\frac{d}{dx} \left( \sin(4x) \right) = 4\cos(4x)
\]

### **Question 41:**
**Equation: \( \int \frac{1}{x^2} \, dx \)**

**Solution:**
The integral of \(\frac{1}{x^2}\) with respect to \(x\) is:
\[
\int \frac{1}{x^2} \, dx = -\frac{1}{x} + C
\]
where \(C\) is the constant of integration.

### **Question 43:**
**Equation: \( \frac{d}{dx} \left( \tan(5x) \right) \)**

**Solution:**
The derivative of \(\tan(5x)\) with respect to \(x\) is:
\[
\frac{d}{dx} \left( \tan(5x) \right) = 5\sec^2(5x)
\]

### **Question 45:**
**Equation: \( \int \frac{1}{2x} \, dx \)**

**Solution:**
The integral of \(\frac{1}{2x}\) with respect to \(x\) is:
\[
\int \frac{1}{2x} \, dx = \frac{1}{2} \ln|x| + C
\]
where \(C\) is the constant of integration.

### **Question 47:**
**Equation: \( \frac{d}{dx} \left( e^{7x} \right) \)**

**Solution:**
The derivative of \(e^{7x}\) with respect to \(x\) is:
\[
\frac{d}{dx} \left( e^{7x} \right) = 7e^{7x}
\]

### **Question 51:**
**Equation: \( \frac{d}{dx} \left( \frac{1}{x^3} \right) \)**

**Solution:**
The derivative of \(\frac{1}{x^3}\) with respect to \(x\) is:
\[
\frac{d}{dx} \left( \frac{1}{x^3} \right) = -\frac{3}{x^4}
\]

### **Question 54:**
**Equation: \( \int \ln(x) \, dx \)**

**Solution:**
The integral of \(\ln(x)\) with respect to \(x\) is:
\[
\int \ln(x) \, dx = x\ln(x) - x + C
\]
where \(C\) is the constant of integration.

---

Would you like further details on any of these solutions, or do you have any specific questions? 

### **Follow-up Questions:**
1. How does the chain rule apply in finding the derivative of composite functions like \(\ln(2x)\)?
2. Why is the constant of integration important in indefinite integrals?
3. Can you provide a step-by-step explanation of the integral of \(\sin(2x)\)?
4. How would you approach finding the derivative of more complex functions like \(e^{x^2}\)?
5. What are some common mistakes to avoid when differentiating or integrating trigonometric functions?

### **Tip:**
When dealing with integrals, always remember to include the constant of integration \(C\) in your final answer, as indefinite integrals represent a family of functions differing by a constant.

Explore detailed solutions for calculus problems including derivatives and integrals of exponential, trigonometric, and logarithmic functions. Understand key concepts like the chain rule and the fundamental theorem of calculus. Suitable for advanced high school students and calculus enthusiasts.

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