Integral of $e^{-9 x}$

Q: What is the integral of exp(-9 * x)?

The integral of e^-9 x is -e^-9 x9 + C.

The integral of $e^{-9 x}$ is $\frac{-e^{-9 x}}{9} + C$ .

\int e^{-9 x}\,dx = \frac{-e^{-9 x}}{9} + C

Explain it step by step in chat Make a short video of it

Graph of f(x) = exp(-9 * x) with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.1110974.

How to work it out

1
We want an antiderivative of f(x) = exp(-9 * x).
$\int e^{-9 x}\,dx$
2
Method: ∫ $e^{−kx}$ dx = − $e^{−kx}$ /k.
3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
$\int e^{-9 x}\,dx = \frac{-e^{-9 x}}{9} + C$
4
Check by differentiating the answer; you must get the original function back:
$\frac{d}{dx}\left[\frac{-e^{-9 x}}{9}\right] = \frac{-\left(\left(-81\right) e^{\left(-9\right) x}\right)}{81}$
5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:
$\int_{0}^{1} e^{-9 x}\,dx = \Big[\frac{-e^{-9 x}}{9}\Big]_{0}^{1} \approx 0.1110974$

LaTeX source of the formulas on this page

\[ \int e^{-9 x}\,dx = \frac{-e^{-9 x}}{9} + C \]
\[ \int e^{-9 x}\,dx \]
\[ \frac{d}{dx}\left[\frac{-e^{-9 x}}{9}\right] = \frac{-\left(\left(-81\right) e^{\left(-9\right) x}\right)}{81} \]
\[ \int_{0}^{1} e^{-9 x}\,dx = \Big[\frac{-e^{-9 x}}{9}\Big]_{0}^{1} \approx 0.1110974 \]

Definite integrals of the same function

Interval	Value
from 0 to 1	0.1110974
from 0 to 0.5	0.10987678

Questions

What is the integral of exp(-9 * x)?

The integral of $e^{-9 x}$ is $\frac{-e^{-9 x}}{9} + C$ .

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(-9 * x) from 0 to 1?

About 0.1110974.

Integral of e−9xe^{-9 x}e−9x

How to work it out

Definite integrals of the same function

Questions

More integrals

Integral of $e^{-9 x}$