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Calculus I · Unit 3A · lesson

Basic Antiderivative Rules

Concept

Learning objectives

Use the standard power, exponential, logarithmic, and trigonometric antiderivative formulas.

Basic Antiderivative Rules

Explanation

Integration rules are derivative rules read backward

The basic antiderivative table is built from familiar derivatives. Powers, exponentials, logarithms, and trigonometric functions are integrated by asking which function differentiates to the given integrand. This reverse-recognition viewpoint is more durable than treating the table as a disconnected list. It also makes verification immediate: differentiate the proposed answer.

Reverse rules require attention to constants and domains. A coefficient may need to be divided out, the power rule excludes exponent −1-1, and ∫1/x dx\int 1/x\,dx requires ln⁡∣x∣\ln|x|. No basic table can replace algebraic simplification. Rewriting radicals and reciprocals as powers often reveals that a seemingly new integral is simply an old rule in less cooperative clothing.

For n≠−1n\ne-1,

∫xn dx=xn+1n+1+C.\int x^n\,dx=\frac{x^{n+1}}{n+1}+C.

Other central formulas include

∫ex dx=ex+C,∫1x dx=ln⁡∣x∣+C,\int e^x\,dx=e^x+C, \qquad \int\frac1x\,dx=\ln|x|+C,∫cos⁡x dx=sin⁡x+C,∫sin⁡x dx=−cos⁡x+C,\int\cos x\,dx=\sin x+C, \qquad \int\sin x\,dx=-\cos x+C,∫sec⁡2x dx=tan⁡x+C,∫11+x2 dx=arctan⁡x+C.\int\sec^2x\,dx=\tan x+C, \qquad \int\frac1{1+x^2}\,dx=\arctan x+C.
Decision

Normalize before integrating

Rewrite roots as fractional powers, move denominator powers into negative exponents when helpful, split sums, and factor out constants. Integration is often easy after the algebra stops wearing a disguise.

Interactive checku3a-basic-rules-01

Evaluate ∫(x3+2sin⁡x) dx\int(x^3+2\sin x)\,dx.

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Show hint

Integrate each term and remember that the antiderivative of sine is negative cosine.

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After the explanation

Use the section idea

Reading lens

Choose an integration method from the integrand's structure, then verify the result by differentiation.

Mental model

Substitution reverses a chain rule, parts reverses a product rule, and algebraic or trigonometric rewrites expose a recognizable derivative pattern.

Decision

Simplify first; look for an inner derivative; then consider parts, identities, trigonometric substitution, or partial fractions in a deliberate order.

Common trap

Choosing a method by surface appearance, transforming only part of the differential, or accepting a more complicated integral than the one you started with.

Check yourself

Can you name the derivative rule being reversed and differentiate the final answer back to the integrand?

Source & rights

Original instruction with traceable references.

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