Integrating the Exponential Function

One short rule integrates any exponential function: carry it down unchanged and divide by ln of its base, giving b^x / ln b + C. Base e is the special case where ln e = 1, so the integral of e^x is e^x itself. A coefficient k in the exponent joins the denominator, and the exponent must stay first degree.

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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 , , or anything else; what matters is where it is.

, , — exponential
, — not exponential, the variable is in the base

The base carries two conditions: it must be positive, and it must not equal . If the base were , then for every — a constant, not an exponential function.

b > 0 \quad\text{and}\quad b \neq 1
Definition The rule

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

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

Note Why base integrates to itself

Apply the same rule with base . The denominator is , and , so the division changes nothing.

The integral of is itself. That is one of the reasons is given its own symbol.

Note What the base controls

With the exponent fixed at , 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 , because for any base.

1 x small base e large base
The base 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 , 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 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 they grow by a fixed amount; under an exponential the amount of the increase is itself increasing.

0 1 2 x y = e x
The strip from to dwarfs the strip from to , 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 of its base.

— since

The base can be less than and the method does not change; is simply a negative number.

Note A coefficient in the exponent

If the exponent is rather than , one thing changes: the coefficient joins the denominator.

— the leaves only the

This holds only while the exponent is first degree. or is a different problem the rule does not cover.

Summary
  1. A function is exponential when the variable is in the exponent, with base b > 0 and .
  2. : keep the function, divide by of the base.
  3. , because .
  4. A larger base climbs faster, and the area it accumulates accelerates rather than growing at a fixed rate.
  5. For an exponent , put the coefficient in the denominator: .
  6. The exponent must be first degree; is outside the rule.
283. Relations and Functions
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