WordGPT

Integral of eāˆ’3xe^{-3 x}

The integral of eāˆ’3xe^{-3 x} is āˆ’eāˆ’3x3+C\frac{-e^{-3 x}}{3} + C.

Explain it step by step in chatMake a short video of it
Graph of f(x) = exp(-3 * 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(-3 * x) with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.31673764.

How to work it out

  1. 1

    We want an antiderivative of f(x) = exp(-3 * x).

    āˆ«eāˆ’3xā€‰dx\int e^{-3 x}\,dx
  2. 2

    Method: āˆ«eāˆ’kxe^{āˆ’kx} dx = āˆ’eāˆ’kxe^{āˆ’kx}/k.

  3. 3

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

    āˆ«eāˆ’3xā€‰dx=āˆ’eāˆ’3x3+C\int e^{-3 x}\,dx = \frac{-e^{-3 x}}{3} + C
  4. 4

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

    ddx[āˆ’eāˆ’3x3]=āˆ’((āˆ’9)e(āˆ’3)x)9\frac{d}{dx}\left[\frac{-e^{-3 x}}{3}\right] = \frac{-\left(\left(-9\right) e^{\left(-3\right) x}\right)}{9}
  5. 5

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

    āˆ«01eāˆ’3xā€‰dx=[āˆ’eāˆ’3x3]01ā‰ˆ0.31673764\int_{0}^{1} e^{-3 x}\,dx = \Big[\frac{-e^{-3 x}}{3}\Big]_{0}^{1} \approx 0.31673764
LaTeX source of the formulas on this page
\[ \int e^{-3 x}\,dx = \frac{-e^{-3 x}}{3} + C \]
\[ \int e^{-3 x}\,dx \]
\[ \frac{d}{dx}\left[\frac{-e^{-3 x}}{3}\right] = \frac{-\left(\left(-9\right) e^{\left(-3\right) x}\right)}{9} \]
\[ \int_{0}^{1} e^{-3 x}\,dx = \Big[\frac{-e^{-3 x}}{3}\Big]_{0}^{1} \approx 0.31673764 \]

Definite integrals of the same function

IntervalValue
from 0 to 10.31673764
from 0 to 0.50.25895661

Questions

What is the integral of exp(-3 * x)?

The integral of eāˆ’3xe^{-3 x} is āˆ’eāˆ’3x3+C\frac{-e^{-3 x}}{3} + 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(-3 * x) from 0 to 1?

About 0.31673764.

More integrals

    Integral of exp(-3 * x): -exp(-3 * x) / 3 + C (with steps) | WordGPT