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Indefinite integralsCalculusCollegeabout 0:56

Integrate (ln x)/x: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: ln(x)²/2 + 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
$\int \frac{\ln\left( x\right)}{x}\,dx$
2. Steps, part 1
1. $\int \frac{\ln\left( x\right)}{x}\,dx$
  Start with the integral
2. $\int \frac{\ln\left( x\right)}{x}\,dx = \int {u}^{1}\,du$
  Substitute u = ln(x), so du = (1 / x) dx
3. $\int {u}^{1}\,du = \frac{{u}^{2}}{2}$
  Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
4. $\int \frac{\ln\left( x\right)}{x}\,dx = \frac{{\ln\left( x\right)}^{2}}{2}$
  Substitute back u = ln(x)
3. Steps, part 2
1. $\int \frac{\ln\left( x\right)}{x}\,dx = \frac{{\ln\left( x\right)}^{2}}{2} + C$
  Simplify and add the constant of integration
4. The answer
$\frac{{\ln\left( x\right)}^{2}}{2} + C$
∫ (ln(x)) / x dx = ln(x)²/2 + C
5. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

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

Steps, part 1. Start with the integral. Substitute u equals ln(x), so du equals (1 / x) dx. Power rule: the integral of x to the power n dx equals x to the power n plus 1/(n plus 1). Substitute back u equals ln(x).

Steps, part 2. Simplify and add the constant of integration.

The answer. So the answer is ln(x) squared/2 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 (ln x)/x: Indefinite integrals Step-by-Step Lesson