Integrating an exponential function is one of the shortest rules in calculus, provided a few details are settled first: what counts as an exponential function, and what its base is allowed to be.

Note What makes a function exponential

In an exponential function the variable sits in the exponent and a fixed number sits in the base. The variable can be $x$ , $t$ , or anything else; what matters is where it is.

a^{x}

2^{x}

e^{x}

— exponential

x^{2}

x^{3}

— not exponential, the variable is in the base

The base carries two conditions: it must be positive, and it must not equal $1$ . If the base were $1$ , then $1^{x} = 1$ for every $x$ — a constant, not an exponential function.

b > 0 \quad\text{and}\quad b \neq 1

Definition The rule

\int b^{x}\,dx = \frac{b^{x}}{\ln b} + C

The function is carried down unchanged. It is divided by $\ln$ of the base — a fixed number you can read off a calculator — and the constant $C$ is added as always. Nothing is expanded or rewritten first.

Keep the function as it is; divide by $\ln$ of the base.

Note Why base

e

integrates to itself

Apply the same rule with base $e$ . The denominator is $\ln e$ , and $\ln e = 1$ , so the division changes nothing.

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

The integral of $e^{x}$ is $e^{x}$ itself. That is one of the reasons $e$ is given its own symbol.

Note What the base controls

With the exponent fixed at $x$ , the base sets how hard the curve climbs. A smaller base rises gently; a larger base rises faster and bends harder. Every curve passes through $(0,\, 1)$ , because $b^{0} = 1$ for any base.

The base

e

sits between the extremes, and its rate of growth is the one that turns up in natural growth and decay.

Because the denominator in the rule is $\ln b$ , a larger base means a larger divisor — the integral is scaled down by exactly the number that measures how fast the curve climbs.

Note Area that speeds up

Sweep left to right under $e^{x}$ one step at a time. Each step adds a strip of area, and every strip is taller than the one before it. For a constant function those strips are equal; for $x$ they grow by a fixed amount; under an exponential the amount of the increase is itself increasing.

The strip from

1

2

dwarfs the strip from

0

1

, and the next one dwarfs that. This is why the integral of an exponential is another exponential rather than a polynomial: only an exponential can keep pace with area that accelerates.

Example Four bases, one method

Carry the function down unchanged, then divide by $\ln$ of its base.

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

— since

\ln e = 1

\displaystyle\int 2^{x}\,dx = \dfrac{2^{x}}{\ln 2} + C

\displaystyle\int 5^{x}\,dx = \dfrac{5^{x}}{\ln 5} + C

\displaystyle\int \left(\tfrac{1}{2}\right)^{x}\,dx = \dfrac{\left(\tfrac{1}{2}\right)^{x}}{\ln \tfrac{1}{2}} + C

The base can be less than $1$ and the method does not change; $\ln \tfrac{1}{2}$ is simply a negative number.

Note A coefficient in the exponent

If the exponent is $kx$ rather than $x$ , one thing changes: the coefficient $k$ joins the denominator.

\int b^{kx}\,dx = \frac{b^{kx}}{k\,\ln b} + C

\displaystyle\int e^{2x}\,dx = \dfrac{e^{2x}}{2} + C

— the

\ln e = 1

leaves only the

2

\displaystyle\int 2^{3x}\,dx = \dfrac{2^{3x}}{3\,\ln 2} + C

\displaystyle\int 10^{2x}\,dx = \dfrac{10^{2x}}{2\,\ln 10} + C

This holds only while the exponent is first degree. $e^{x^{2}}$ or $2^{x^{3}}$ is a different problem the rule does not cover.

Summary

A function is exponential when the variable is in the exponent, with base $b > 0$ and $b \neq 1$ .
$\displaystyle\int b^{x}\,dx = \frac{b^{x}}{\ln b} + C$ : keep the function, divide by $\ln$ of the base.
$\displaystyle\int e^{x}\,dx = e^{x} + C$ , because $\ln e = 1$ .
A larger base climbs faster, and the area it accumulates accelerates rather than growing at a fixed rate.
For an exponent $kx$ , put the coefficient $k$ in the denominator: $\displaystyle\int b^{kx}\,dx = \frac{b^{kx}}{k\,\ln b} + C$ .
The exponent must be first degree; $e^{x^{2}}$ is outside the rule.

Integrating the Exponential Function