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Indefinite integralsCalculusCollegeabout 1:13

Antiderivative of (x + 1)/x: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: ln(abs(x)) + x + C.

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 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Evaluating an integral
    ∫x+1x dx\int \frac{x+1}{x}\,dx
  2. 2. Steps, part 1
    1. ∫x+1x dx\int \frac{x+1}{x}\,dx
      Start with the integral
    2. x+1x=1x+1\frac{x+1}{x} = \frac{1}{x}+1
      Divide each term of the numerator by the denominator
    3. ∫x−1 dx=ln⁡(∣x∣)\int {x}^{-1}\,dx = \ln\left(\left| x\right|\right)
      Reciprocal rule: ∫ 1/x dx = ln|x|
    4. ∫1x−1 dx=ln⁡(∣x∣)\int 1{x}^{-1}\,dx = \ln\left(\left| x\right|\right)
      Constant multiple rule: ∫ k·f dx = k∫ f dx
  3. 3. Steps, part 2
    1. ∫x0 dx=x\int {x}^{0}\,dx = x
      Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
    2. ∫1x0 dx=x\int 1{x}^{0}\,dx = x
      Constant multiple rule: ∫ k·f dx = k∫ f dx
    3. ∫1x−1+1x0 dx=ln⁡(∣x∣)+x\int 1{x}^{-1}+1{x}^{0}\,dx = \ln\left(\left| x\right|\right)+ x
      Sum rule: integrate term by term
    4. ∫x+1x dx=ln⁡(∣x∣)+x+C\int \frac{x+1}{x}\,dx = \ln\left(\left| x\right|\right)+ x + C
      Simplify and add the constant of integration
  4. 4. The answer
    ln⁡(∣x∣)+x+C\ln\left(\left| x\right|\right)+ x + C

    ∫ (x + 1) / x dx = ln(abs(x)) + x + C

  5. 5. Check
    • Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

The problem. Here is the problem: integral of (x plus 1)/x. Find the indefinite integral (antiderivative) of (x plus 1) / x with respect to x.

Steps, part 1. Start with the integral. Divide each term of the numerator by the denominator. Reciprocal rule: the integral of 1/x dx equals ln|x|. Constant multiple rule: the integral of k times f dx equals k the integral of f dx.

Steps, part 2. Power rule: the integral of x to the power n dx equals x to the power n plus 1/(n plus 1). Constant multiple rule: the integral of k times f dx equals k the integral of f dx. Sum rule: integrate term by term. Simplify and add the constant of integration.

The answer. So the answer is ln(abs(x)) plus x 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.

More lessons like this

Antiderivative of (x + 1)/x: Animated Step-by-Step Lesson