Integral of eᵏˣ: formula, calculator and worked example

What the symbols mean

$I$: the value of $∫ₐ^bAe^(kx) dx$
$A$: the coefficient
$k$: the rate (if 0, the integrand is constant)
$a$: the lower limit
$b$: the upper limit
$C$: the constant of integration

When to use it

Use it for total amounts from exponential rates: total growth over time, total drug absorbed, the area under a decay curve. With k < 0 and a large upper limit it shows why the total of a decaying process levels off.

Worked example

Find the area under y = eˣ from x = 0 to x = 1 (A = 1, k = 1, a = 0, b = 1).

The antiderivative of $e^{k x}$ is $\frac{1}{k} e^{k x}$ (differentiate it to check: the chain rule brings back the k).
$\int_{a}^{b} A e^{k x} d x = \frac{A (e^{k b} - e^{k a})}{k}$
Substitute and work it out:
$I = \frac{1 (e^{1 \cdot 1} - e^{1 \cdot 0})}{1}$
$= \frac{1 (e^{1} - e^{0})}{1}$
$= \frac{1 (e^{1} - 1)}{1}$
$\approx 1.71828$
Numerical check: Simpson's rule with 400 strips gives $1.718281828$ .

Answer I = e − 1 ≈ 1.7183

Why it works

Differentiating $1/ke^(kx)$ with the chain rule gives $1/k·ke^(kx) = e^(kx)$ .

Common mistakes

Multiplying by k instead of dividing: $∫e^(3x)dx = 1/3e^(3x)$ .
Forgetting that $e⁰ = 1$ when substituting a lower limit of 0, not 0.
Using the power rule on $e^x$ .

Integral of eᵏˣ

Calculator

See it move

What the symbols mean

When to use it

Worked example

Why it works

Common mistakes