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Indefinite integralsCalculusCollegeabout 0:48

Antiderivative of e^(-x): animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: -(1 / e^x) + 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
$\int {e}^{- x}\,dx$
2. Step by step
1. $\int {e}^{- x}\,dx$
  Start with the integral
2. $\int {e}^{- x}\,dx = -\frac{1}{{e}^{x}}$
  Exponential rule: ∫ eᵘ du = eᵘ (divide by the inner slope)
3. $\int {e}^{- x}\,dx = -\frac{1}{{e}^{x}} + C$
  Simplify and add the constant of integration
3. The answer
$-\frac{1}{{e}^{x}} + C$
∫ e^(-x) dx = -(1 / e^x) + 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 e to the power negative x. Find the indefinite integral (antiderivative) of e to the power negative x with respect to x.

Step by step. Start with the integral. Exponential rule: the integral of eᵘ du equals eᵘ (divide by the inner slope). Simplify and add the constant of integration.

The answer. So the answer is -(1 / e to the power 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.

Antiderivative of e^(-x): Animated Step-by-Step Lesson