Evaluate the following integral using integration by parts.

Integral from nothing to nothing x squared ln x dx

To evaluate the integral 

\[
\int x^2 \ln(x) \, dx,
\]

we can use the technique of integration by parts. The formula for integration by parts is:

\[
\int u \, dv = uv - \int v \, du.
\]

### Step 1: Choose \( u \) and \( dv \)

Let's choose:
\[
u = \ln(x), \quad dv = x^2 \, dx.
\]

Then we compute \( du \) and \( v \):

\[
du = \frac{1}{x} \, dx, \quad v = \int x^2 \, dx.
\]

To find \( v \), integrate \( x^2 \):

\[
v = \int x^2 \, dx = \frac{x^3}{3}.
\]

### Step 2: Apply the Integration by Parts Formula

Substitute into the integration by parts formula:

\[
\int x^2 \ln(x) \, dx = u v - \int v \, du.
\]

Substitute the expressions for \( u, v, du, \) and \( dv \):

\[
\int x^2 \ln(x) \, dx = \ln(x) \cdot \frac{x^3}{3} - \int \frac{x^3}{3} \cdot \frac{1}{x} \, dx.
\]

Simplify the expression:

\[
\int x^2 \ln(x) \, dx = \frac{x^3 \ln(x)}{3} - \int \frac{x^2}{3} \, dx.
\]

\[
\int x^2 \ln(x) \, dx = \frac{x^3 \ln(x)}{3} - \frac{1}{3} \int x^2 \, dx.
\]

### Step 3: Evaluate the Remaining Integral

The remaining integral is \( \int x^2 \, dx \):

\[
\int x^2 \, dx = \frac{x^3}{3}.
\]

Thus,

\[
\frac{1}{3} \int x^2 \, dx = \frac{1}{3} \cdot \frac{x^3}{3} = \frac{x^3}{9}.
\]

### Step 4: Combine Results

Substitute this back into the expression:

\[
\int x^2 \ln(x) \, dx = \frac{x^3 \ln(x)}{3} - \frac{x^3}{9} + C.
\]

### Final Answer

\[
\int x^2 \ln(x) \, dx = \frac{x^3 \ln(x)}{3} - \frac{x^3}{9} + C.
\]

This is the evaluated integral using integration by parts.

Would you like more details on any step or have any other questions?

Here are 5 related questions for further practice:

1. Evaluate \(\int x \ln(x) \, dx\) using integration by parts.
2. Solve \(\int x^2 e^x \, dx\) using integration by parts.
3. Calculate \(\int \ln(x) \, dx\) using integration by parts.
4. Determine \(\int x^3 \ln(x) \, dx\) using integration by parts.
5. Evaluate \(\int x \ln(x^2) \, dx\) using integration by parts.

**Tip:** When choosing \(u\) and \(dv\) for integration by parts, pick \(u\) to be a function that simplifies upon differentiation, and \(dv\) such that it is straightforward to integrate.

Learn how to evaluate the integral ∫ x^2 ln(x) dx using integration by parts. Step-by-step solution with detailed computation and explanation. Suitable for college-level students and calculus enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation