To solve the integral \(\int \left(\ln(x)\right)^2 \, dx\), we can use integration by parts. Let's go through the solution step by step.

### Step 1: Set up integration by parts
We know the formula for integration by parts:
\[
\int u \, dv = uv - \int v \, du
\]
Here, let's choose:
\[
u = (\ln(x))^2 \quad \text{and} \quad dv = dx
\]

### Step 2: Differentiate \(u\) and integrate \(dv\)
First, differentiate \(u\):
\[
du = 2\ln(x) \cdot \frac{1}{x} \, dx = \frac{2\ln(x)}{x} \, dx
\]
Next, integrate \(dv\):
\[
v = \int dx = x
\]

### Step 3: Apply the integration by parts formula
Now, substitute into the integration by parts formula:
\[
\int (\ln(x))^2 \, dx = x(\ln(x))^2 - \int x \cdot \frac{2\ln(x)}{x} \, dx
\]
This simplifies to:
\[
\int (\ln(x))^2 \, dx = x(\ln(x))^2 - 2\int \ln(x) \, dx
\]

### Step 4: Solve \(\int \ln(x) \, dx\)
We now need to solve the integral \(\int \ln(x) \, dx\). We can again use integration by parts, with:
\[
u = \ln(x) \quad \text{and} \quad dv = dx
\]
Differentiate and integrate as before:
\[
du = \frac{1}{x} \, dx \quad \text{and} \quad v = x
\]
Apply the integration by parts formula:
\[
\int \ln(x) \, dx = x\ln(x) - \int x \cdot \frac{1}{x} \, dx = x\ln(x) - \int 1 \, dx = x\ln(x) - x
\]

### Step 5: Substitute back into the original equation
Substitute the result of \(\int \ln(x) \, dx\) back into the original equation:
\[
\int (\ln(x))^2 \, dx = x(\ln(x))^2 - 2\left(x\ln(x) - x\right) + C
\]
Simplify the expression:
\[
\int (\ln(x))^2 \, dx = x(\ln(x))^2 - 2x\ln(x) + 2x + C
\]

Thus, the final answer is:
\[
\boxed{\int (\ln(x))^2 \, dx = x(\ln(x))^2 - 2x\ln(x) + 2x + C}
\]

Would you like any further details or explanations? Here are some additional related questions to explore:

1. How would you apply integration by parts to \(\int x \ln(x) \, dx\)?
2. What is the integral of \(\int \ln(x) \, dx\) if you change the variable \(x\) to \(e^t\)?
3. How does the integral change if \((\ln(x))^2\) is replaced with \((\ln(x))^3\)?
4. Can you apply a substitution method to solve \(\int \left(\ln(x)\right)^2 \, dx\)?
5. What are the general steps for choosing \(u\) and \(dv\) in integration by parts?

**Tip:** Always double-check the constants and coefficients when using integration by parts, as small errors can propagate through the calculation.

Explore the integral of (ln(x))^2 with a detailed step-by-step solution using integration by parts. Understand the application of key formulas in integral calculus. Suitable for advanced high school students and anyone interested in calculus problems.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation