Calculus — Integrals⏱ 6 min read

Basic Integration Rules

∫ eˣ dx = eˣ + C

What is the Basic Integration Rules?

Every indefinite integral includes "+ C", the constant of integration, because differentiation destroys constant terms — so integration (its reverse) can never recover what that constant originally was. C represents that unknown value: x² + 5 and x² − 100 have exactly the same derivative, 2x, so integrating 2x can only ever recover x² plus some unknown constant, not the specific constant either original function happened to have.

The result ∫(1/x)dx = ln|x| + C fills a gap left by the power rule for integration: the power rule ∫xⁿdx = xⁿ⁺¹/(n+1) + C fails exactly at n = −1, since that would require dividing by zero. The logarithm is the special-case solution for that one exponent, and it's a common early mistake to try applying the power rule there anyway.

These basic integrals are the ones every other integration technique — substitution, integration by parts, partial fractions — eventually reduces down to. Learning to recognize when a more complicated-looking integral is secretly one of these in disguise (often after a substitution) is most of what makes integration feel manageable rather than like guesswork.

What Each Variable Means

Constant of integrationAdded to every indefinite integral, since the derivative of any constant is zero — infinitely many functions share the same derivative.

Constant or baseDepending on context: a constant multiplier, or the base of an exponential.

When to Use It

Finding the antiderivative of a constant, 1/x, eˣ, aˣ, or ln x
As building blocks for more complex integrals via substitution or integration by parts
Evaluating definite integrals of these basic function types

Step-by-Step Examples

Example 1: A definite integral

Problem: Evaluate ∫₁³ (2/x) dx

Find the antiderivative

Apply the 1/x rule, with the constant 2 carried through.

∫ 2/x dx = 2 ln|x| + C

Apply the limits [1, 3]

Evaluate the antiderivative at both bounds and subtract.

[2 ln|x|]₁³ = 2 ln 3 − 2 ln 1

Simplify using ln 1 = 0

The lower bound term vanishes.

= 2 ln 3

✓

Answer: 2 ln 3 ≈ 2.197

Example 2: Deriving the ln x integral

Problem: Show that ∫ ln x dx = x ln x − x + C, using integration by parts (∫u dv = uv − ∫v du)

Choose u and dv

Let u = ln x and dv = dx.

du = (1/x)dx, v = x

Apply the integration by parts formula

Substitute into uv − ∫v du.

x ln x − ∫ x·(1/x) dx = x ln x − ∫ 1 dx

✓

Answer: ∫ ln x dx = x ln x − x + C

Common Mistakes

Mistake: Forgetting the + C on an indefinite integral.
Fix: Every indefinite integral needs + C, since differentiating any constant gives zero — without it, you're only reporting one specific antiderivative out of infinitely many.
Mistake: Trying to use the power rule on ∫(1/x)dx.
Fix: The power rule ∫xⁿdx = xⁿ⁺¹/(n+1) + C breaks down at n = -1 (division by zero) — that specific case uses ln|x| + C instead.

Practice Questions

Evaluate ∫ eˣ dx.
Evaluate ∫₁ᵉ (1/x) dx.
Hint: The antiderivative is ln|x|; ln(e) − ln(1) = 1 − 0.

Frequently Asked Questions

Why does ∫1/x dx use absolute value bars?

Because ln x is only defined for positive x, but 1/x is defined for all x ≠ 0. Using ln|x| extends the antiderivative to negative x as well.

What's the integral of a constant a by itself?

∫a dx = ax + C — since the derivative of ax is simply a.

Related Formulas

Calculus — Derivatives

d/dx(eˣ) = eˣ

Exponential & Log Derivatives

The three core derivatives involving exponential and logarithmic functions: eˣ, aˣ, and ln x.

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Calculus — Integrals

∫ sin x dx = −cos x + C

Trigonometric Integrals

The complete table of ten standard trigonometric integrals — each the reverse of a derivative rule.

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Calculus — Integrals

∫ dx/√(a²−x²) = arcsin(x/a) + C

Special Integrals

Two integral forms that arise from square roots of quadratic expressions, producing inverse-trig or logarithmic results.

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