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Integral of exe^{x}

The integral of exe^{x} is ex+Ce^{x} + C.

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Graph of f(x) = exp(x) with the area from 0 to 1 shaded-0.4-0.20.20.40.60.811.21.41.612345f(x)
Graph of f(x) = exp(x) with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 1.7182818.

How to work it out

  1. 1

    We want an antiderivative of f(x) = exp(x).

    āˆ«exā€‰dx\int e^{x}\,dx
  2. 2

    Method: standard integral.

  3. 3

    That gives the antiderivative; add the constant C because any constant disappears when you differentiate:

    āˆ«exā€‰dx=ex+C\int e^{x}\,dx = e^{x} + C
  4. 4

    Check by differentiating the answer; you must get the original function back:

    ddx[ex]=ex\frac{d}{dx}\left[e^{x}\right] = e^{x}
  5. 5

    As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:

    āˆ«01exā€‰dx=[ex]01ā‰ˆ1.7182818\int_{0}^{1} e^{x}\,dx = \Big[e^{x}\Big]_{0}^{1} \approx 1.7182818
LaTeX source of the formulas on this page
\[ \int e^{x}\,dx = e^{x} + C \]
\[ \int e^{x}\,dx \]
\[ \frac{d}{dx}\left[e^{x}\right] = e^{x} \]
\[ \int_{0}^{1} e^{x}\,dx = \Big[e^{x}\Big]_{0}^{1} \approx 1.7182818 \]

Definite integrals of the same function

IntervalValue
from 0 to 11.7182818
from 0 to 0.50.64872127

Questions

What is the integral of exp(x)?

The integral of exe^{x} is ex+Ce^{x} + C.

Why do we add + C?

Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.

What is the integral of exp(x) from 0 to 1?

About 1.7182818.

More integrals

    Integral of exp(x): exp(x) + C (with steps) | WordGPT