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Indefinite integralsCalculusCollegeabout 0:44

Antiderivative of ln(x): animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: x(ln(x) - 1) + 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
$\int \ln\left( x\right)\,dx$
2. Step by step
1. $\int \ln\left( x\right)\,dx$
  Start with the integral
2. $\int \ln\left( x\right)\,dx = x \left(\ln\left( x\right)-1\right)$
  ∫ ln x dx = x ln x − x (integration by parts)
3. $\int \ln\left( x\right)\,dx = x \left(\ln\left( x\right)-1\right) + C$
  Simplify and add the constant of integration
3. The answer
$x \left(\ln\left( x\right)-1\right) + C$
∫ ln(x) dx = x(ln(x) - 1) + C
4. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

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

Step by step. Start with the integral. the integral of ln x dx equals x ln x minus x (integration by parts). Simplify and add the constant of integration.

The answer. So the answer is x(ln(x) minus 1) 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.

Antiderivative of ln(x): Animated Step-by-Step Lesson