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Indefinite integralsCalculusCollegeabout 0:56

Evaluate ∫ x e^x dx: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: e^x·(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 (0.4 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int x {e}^{x}\,dx$
2. Step by step
1. $\int x {e}^{x}\,dx$
  Start with the integral
2. $\int u\,dv = uv - \int v\,du$
  Integration by parts (tabular method)
3. $\int x {e}^{x}\,dx = {e}^{x} \left( x-1\right)$
  Combine the alternating products
4. $\int x {e}^{x}\,dx = {e}^{x} \left( x-1\right) + C$
  Simplify and add the constant of integration
3. The answer
${e}^{x} \left( x-1\right) + C$
∫ x e^x dx = e^x·(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: the integral of x e to the power x dx. Find the indefinite integral (antiderivative) of x e to the power x with respect to x.

Step by step. Start with the integral. Integration by parts (tabular method). Differentiate the polynomial x until it reaches 0, integrate e to the power x the same number of times, and combine with alternating signs. Combine the alternating products. Simplify and add the constant of integration.

The answer. So the answer is e to the power x times (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.

Evaluate ∫ x e^x dx: Animated Step-by-Step Lesson (College)