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Indefinite integralsCalculusCollegeabout 0:55

Integrate 1/((x+1)(x+4)): animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: -(ln(abs(x + 4)) / 3) + ln(abs(x + 1)) / 3 + 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 (0.4 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int \frac{1}{\left( x+1\right) \left( x+4\right)}\,dx$
2. Step by step
1. $\int \frac{1}{\left( x+1\right) \left( x+4\right)}\,dx$
  Start with the integral
2. $\int \frac{1}{\left( x+1\right) \left( x+4\right)}\,dx = -\frac{\ln\left(\left| x+4\right|\right)}{3}+\frac{\ln\left(\left| x+1\right|\right)}{3}$
  Partial fractions: split 1/((x − r₁)(x − r₂)) into A/(x − r₁) + B/(x − r₂)
3. $\int \frac{1}{\left( x+1\right) \left( x+4\right)}\,dx = -\frac{\ln\left(\left| x+4\right|\right)}{3}+\frac{\ln\left(\left| x+1\right|\right)}{3} + C$
  Simplify and add the constant of integration
3. The answer
$-\frac{\ln\left(\left| x+4\right|\right)}{3}+\frac{\ln\left(\left| x+1\right|\right)}{3} + C$
∫ 1 / ((x + 1) (x + 4)) dx = -(ln(abs(x + 4)) / 3) + ln(abs(x + 1)) / 3 + C
4. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

The problem. Here is the problem: integrate 1/((x plus 1)(x plus 4)). Find the indefinite integral (antiderivative) of 1 / ((x plus 1) (x plus 4)) with respect to x.

Step by step. Start with the integral. Partial fractions: split 1/((x minus r₁)(x minus r₂)) into A/(x minus r₁) plus B/(x minus r₂). Simplify and add the constant of integration.

The answer. So the answer is -(ln(abs(x plus 4)) / 3) plus ln(abs(x plus 1)) / 3 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 1/((x+1)(x+4)): Animated Step-by-Step Lesson