Integral of e^x: Antidifferentiate the natural exponential in one step

On this page: The Principle | Conditions | Failure Modes | EE Questions | Retrieval Practice | Practice Ground | Solve a Problem | FAQ | How This Fits

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The Principle

The move: Integrate the exact local pattern $e^x$ in one step by copying the function and appending $+C$.

The invariant: This produces an antiderivative family whose derivative returns $e^x$ exactly.

Pattern: $\int e^x\,dx \quad \longrightarrow \quad e^x + C$

Applies directly ✓	Does not apply directly ✗
$\int e^x\,dx \to e^x + C$	$\int e^{2x}\,dx \not\to e^{2x} + C$

Left: this step is a direct match for the rule. Right: the canonical condition still says “always applies,” but this guide is not the governing rule for the whole step because the exponent has extra structure.

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Conditions of Applicability

Condition: always applies

Before applying, check: is this step a direct match for $\int e^x\,dx$, or are you looking at a larger structure such as $e^{g(x)}$, a sum, or a constant multiple?

• The canonical condition is still “always applies.” The real decision is scope: this rule applies directly to the step $\int e^x\,dx$, not automatically to every larger expression that contains base $e$.
• If the exponent is another function, such as in $\int e^{2x}\,dx$, the issue is not a failed condition. The issue is that another method must handle the whole integral.
• In mixed integrals, linearity rules such as the integral sum rule or integral constant multiple rule may govern the outer step while this rule applies to the $e^x$ term inside that larger chain.

Want the complete framework behind this guide? Read Masterful Learning.

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Common Failure Modes

Failure mode: treat every exponential with base $e$ as though this rule governs the whole step -> write $\int e^{g(x)}\,dx = e^{g(x)} + C$ and lose the extra structure that another method must handle.

Debug: ask whether this is a direct $\int e^x\,dx$ match or a larger expression. If the exponent is anything other than the bare variable $x$, stop and identify the governing rule before writing an antiderivative.

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Elaborative Encoding

Use these questions to build deep understanding. (See Elaborative Encoding for the full method.)

Within the Principle

• Why does copying $e^x$ and adding $+C$ produce a correct antiderivative family instead of only one antiderivative?
• What does it mean to say that the derivative of the result returns the original integrand exactly?

For the Principle

• How do you decide whether a step is a direct match for $\int e^x\,dx$ or a larger expression that needs a different outer rule first?
• Why does “always applies” describe the canonical condition without making this the governing rule for every exponential integrand with base $e$?

Between Principles

• How does this rule connect to the derivative of e^x, and why do the derivative and integral versions both preserve the same functional form only for the exact pattern $e^x$?

Generate an Example

• Create one integral where this rule applies directly and one near-miss where it does not, then explain what structural check separates them.

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Retrieval Practice

Answer from memory, then click to reveal and check. (See Retrieval Practice for the full method.)

State the move in one sentence: _____Integrate the exact pattern e^x by copying the function and adding C.

Write the canonical equation: _____$\int e^x\,dx = e^x + C$

State the canonical condition: _____always applies

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Practice Ground

Use these exercises to build move-selection fluency. (See Self-Explanation for how to use worked examples effectively.)

Procedure Walkthrough

Starting from $\int (e^x + x^2)\,dx$, reach a simplified antiderivative while naming which local step uses the integral of $e^x$ rule.

Step	Expression	Operation
0	$\int (e^x + x^2)\,dx$	-
1	$\int e^x\,dx + \int x^2\,dx$	Integral sum rule: split the sum into two integrals
2	$e^x + \int x^2\,dx$	Integral of $e^x$ rule on the exact local pattern $e^x$
3	$e^x + \frac{x^3}{3} + C$	Power rule on $x^2$, then combine the constants of integration

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Drills

Forward step (Format A)

Apply the rule to this direct-match integral.

$\int e^x\,dx$

Reveal

This is a direct match for the rule, so the antiderivative is:

$\int e^x\,dx = e^x + C$

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Apply the correct rule sequence and simplify.

$\int -5e^x\,dx$

Reveal

Use the integral constant multiple rule first, then apply the integral of $e^x$ rule to the $e^x$ term:

$\int -5e^x\,dx = -5\int e^x\,dx = -5e^x + C$

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Does this rule apply directly to the whole step? Explain before writing anything else.

$\int e^{3x}\,dx$

Reveal

No. The canonical condition still says “always applies,” but this is not a direct match for the whole step.

The tempting move

$\int e^{3x}\,dx \not\to e^{3x} + C$

does not apply directly because it ignores the structure in the exponent. This guide covers the direct step $\int e^x\,dx$, not every larger exponential integral.

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Which items contain a direct or local use of the integral of $e^x$ rule in the solution chain?

(i) $\int e^x\,dx$ \quad (ii) $\int e^{x^2}\,dx$ \quad (iii) $\int 4e^x\,dx$ \quad (iv) $\int \frac{1}{x}\,dx$

Reveal

(i) and (iii).

• (i) is a direct match for the rule.
• (iii) uses the integral of $e^x$ rule locally after factoring out the constant: $\int 4e^x\,dx = 4\int e^x\,dx$.
• (ii) is a near-miss because the exponent is not the bare variable $x$.
• (iv) belongs to the integral of 1/x family, not to the exponential rule.

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Action label (Format B)

What was done between these two steps?

$\int e^x\,dx \quad \longrightarrow \quad e^x + C$

Reveal

Integral of $e^x$ rule applied directly. The antiderivative copies the same exponential function and adds the constant of integration.

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What was attempted between these two steps, and why does it not apply directly?

$\int e^{2x}\,dx \quad \longrightarrow \quad e^{2x} + C$

Reveal

An overextended direct use of the integral of $e^x$ rule was attempted.

The issue is not the canonical condition. The issue is scope: the whole step is not a direct $\int e^x\,dx$ match, so the move drops the extra structure in the exponent.

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Name the rule combination used in this completed step.

$\int (3e^x + x^4)\,dx \quad \longrightarrow \quad 3e^x + \frac{x^5}{5} + C$

Reveal

The sum rule split the integral, the constant multiple rule pulled out $3$, the integral of $e^x$ rule handled the $e^x$ term, and the power rule for integrals handled $x^4$.

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Transition identification (Format C)

Which transition uses the integral of $e^x$ rule directly?

$\int (x^2 + e^x)\,dx \xrightarrow{(1)} \int x^2\,dx + \int e^x\,dx \xrightarrow{(2)} \frac{x^3}{3} + \int e^x\,dx \xrightarrow{(3)} \frac{x^3}{3} + e^x + C$

Reveal

Transition (3) uses the integral of $e^x$ rule directly.

• (1) is the sum rule.
• (2) is the power rule on $x^2$.
• (3) is the exact local step $\int e^x\,dx = e^x + C$.

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Which transition does not apply directly, and why?

$\int e^{2x}\,dx \xrightarrow{(1)} e^{2x} + C \xrightarrow{(2)} \text{stop}$

Reveal

Transition (1) does not apply directly.

The move incorrectly treats $e^{2x}$ as though it were a direct $e^x$ match. This guide does not govern that whole-step rewrite.

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Goal micro-chain (Format D)

Starting from $\int (e^x + x^{-1})\,dx$, reach a correct final antiderivative by using the integral of $e^x$ rule only where it fits.

Reveal

First split the sum:

$\int (e^x + x^{-1})\,dx = \int e^x\,dx + \int x^{-1}\,dx$

Apply the integral of $e^x$ rule to the first term:

$\int e^x\,dx = e^x + C_1$

Use the logarithm antiderivative on the second term:

$\int x^{-1}\,dx = \ln\left|x\right| + C_2$

Combine constants:

$\int (e^x + x^{-1})\,dx = e^x + \ln\left|x\right| + C$

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Starting from $\int (2e^x - 3x^2)\,dx$, reach a correct final antiderivative in the minimum number of moves.

Reveal

Split the integral and move constants out:

$\int (2e^x - 3x^2)\,dx = 2\int e^x\,dx - 3\int x^2\,dx$

Apply the local antiderivative rules:

$2\int e^x\,dx - 3\int x^2\,dx = 2e^x - 3\left(\frac{x^3}{3}\right) + C$

Simplify:

$\int (2e^x - 3x^2)\,dx = 2e^x - x^3 + C$

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Solve a Problem

Apply what you’ve learned with Problem Solving.

Problem: Integrate $5e^x + x^3$ and simplify the result.

Full solution

Step	Expression	Move
0	$\int (5e^x + x^3)\,dx$	-
1	$\int 5e^x\,dx + \int x^3\,dx$	Integral sum rule
2	$5\int e^x\,dx + \int x^3\,dx$	Integral constant multiple rule
3	$5e^x + \int x^3\,dx$	Integral of $e^x$ rule on the exact local pattern $e^x$
4	$5e^x + \frac{x^4}{4} + C$	Power rule on $x^3$, then combine constants

Check: differentiating $5e^x + \frac{x^4}{4} + C$ returns $5e^x + x^3$.

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Related Principles

Principle	Relationship
Indefinite integral as antiderivative	Explains why every correct antiderivative family ends with $+C$
Derivative of e^x	Gives the inverse local fact: $e^x$ keeps the same functional form under differentiation
Power rule for integrals	Handles another direct antiderivative pattern often paired with $e^x$ in mixed integrals

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FAQ

What is the integral of $e^x$?

The integral of $e^x$ is $e^x + C$. The function keeps the same form because differentiating $e^x$ returns $e^x$ again.

Why does the condition say “always applies” if I still have to be careful?

The canonical condition says there is no extra algebraic guard on the rule itself. The caution comes from scope: many larger expressions contain base $e$ without making this the governing rule for the whole step.

Can I use this rule on $\int e^{2x}\,dx$?

Not as a one-step whole-rule move. The issue is not a failed condition. The issue is that this guide is not the governing rule for the whole integral when the exponent has extra structure.

Does the rule still work inside a longer integral like $\int (e^x + x^5)\,dx$?

Yes. Use outer linearity rules such as the integral sum rule, then apply the integral of $e^x$ rule to the $e^x$ term inside that larger chain.

Why do I still add $+C$ if the antiderivative looks obvious?

Indefinite integration returns a family of antiderivatives, not one single function. The constant accounts for every vertical shift with the same derivative.

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How This Fits in Unisium

In Unisium, direct integration rules are trained as pattern recognition plus rule selection, not as isolated flashcards. This guide pairs the exact $e^x$ antiderivative fact with contrast cases so you can separate direct matches from near-misses, then stabilize that distinction with retrieval practice, self-explanation, and elaborative encoding. To extend the same workflow across your wider calculus stack, use Masterful Learning or practice directly in the Unisium app.

Explore further:

• Integral sum rule - Split mixed integrals before choosing local antiderivative rules
• Power rule for integrals - Pair the $e^x$ rule with another high-frequency direct antiderivative pattern
• Integral of a^x - Compare the special base $e$ with the general constant-base exponential antiderivative
• Derivative of e^x - Connect the antiderivative fact to its inverse derivative rule

Masterful Learning

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