Integral of $e^{4 x}$

Q: What is the integral of exp(4x)?

The integral of e^4 x is e^4 x4 + C.

The integral of $e^{4 x}$ is $\frac{e^{4 x}}{4} + C$ .

\int e^{4 x}\,dx = \frac{e^{4 x}}{4} + C

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Graph of f(x) = exp(4x) with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 13.399538.

How to work it out

1
We want an antiderivative of f(x) = exp(4x).
$\int e^{4 x}\,dx$
2
Method: ∫ $e^{kx}$ dx = $e^{kx}$ /k.
3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
$\int e^{4 x}\,dx = \frac{e^{4 x}}{4} + C$
4
Check by differentiating the answer; you must get the original function back:
$\frac{d}{dx}\left[\frac{e^{4 x}}{4}\right] = \frac{16 e^{4 x}}{16}$
5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:
$\int_{0}^{1} e^{4 x}\,dx = \Big[\frac{e^{4 x}}{4}\Big]_{0}^{1} \approx 13.399538$

LaTeX source of the formulas on this page

\[ \int e^{4 x}\,dx = \frac{e^{4 x}}{4} + C \]
\[ \int e^{4 x}\,dx \]
\[ \frac{d}{dx}\left[\frac{e^{4 x}}{4}\right] = \frac{16 e^{4 x}}{16} \]
\[ \int_{0}^{1} e^{4 x}\,dx = \Big[\frac{e^{4 x}}{4}\Big]_{0}^{1} \approx 13.399538 \]

Definite integrals of the same function

Interval	Value
from 0 to 1	13.399538
from 0 to 0.5	1.597264

Questions

What is the integral of exp(4x)?

The integral of $e^{4 x}$ is $\frac{e^{4 x}}{4} + C$ .

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(4x) from 0 to 1?

About 13.399538.

Integral of e4xe^{4 x}e4x

How to work it out

Definite integrals of the same function

Questions

More integrals

Integral of $e^{4 x}$