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Indefinite integralsCalculusCollegeabout 0:49

Integrate x^2 ln(x): animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: x³(ln(x) / 3 - 1/9) + C.

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Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.3 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int {x}^{2} \ln\left( x\right)\,dx$
2. Step by step
1. $\int {x}^{2} \ln\left( x\right)\,dx$
  Start with the integral
2. $\int u\,dv = uv - \int v\,du,\quad u = \ln\left( x\right),\ dv = {x}^{2}\,dx$
  Integration by parts with u = ln, dv = polynomial
3. $\int {x}^{2} \ln\left( x\right)\,dx = {x}^{3} \left(\frac{\ln\left( x\right)}{3}-\frac{1}{9}\right)$
  Apply by parts and integrate the remaining polynomial
4. $\int {x}^{2} \ln\left( x\right)\,dx = {x}^{3} \left(\frac{\ln\left( x\right)}{3}-\frac{1}{9}\right) + C$
  Simplify and add the constant of integration
3. The answer
${x}^{3} \left(\frac{\ln\left( x\right)}{3}-\frac{1}{9}\right) + C$
∫ x² ln(x) dx = x³(ln(x) / 3 - 1/9) + C
4. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

The problem. Here is the problem: integrate x squared ln(x). Find the indefinite integral (antiderivative) of x squared ln(x) with respect to x.

Step by step. Start with the integral. Integration by parts with u equals ln, dv equals polynomial. Apply by parts and integrate the remaining polynomial. Simplify and add the constant of integration.

The answer. So the answer is x cubed(ln(x) / 3 minus 1 over 9) plus C. An antiderivative F satisfies F prime of x equals f(x). Because the derivative of any constant is 0, every antiderivative differs by a constant C.

Check. Differentiated the answer and compared it with the integrand at 8 sample points, which checks out.

About indefinite integrals

Find antiderivatives with standard rules, substitution and integration by parts.

Integrate x^2 ln(x): Animated Step-by-Step Lesson (College)