Calculus I · Unit 3A · lesson

Basic Antiderivative Rules

Learn Basic Antiderivative Rules through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.

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Section overview

Computing integrals

What this section is building

Learn Basic Antiderivative Rules through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.

Notice

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

Decide

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

Avoid

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

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 $-1$ , and $\int 1/x\,dx$ requires $\ln|x|$ . 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 $n\ne-1$ ,

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

Other central formulas include

\int e^x\,dx=e^x+C, \qquad \int\frac1x\,dx=\ln|x|+C,

\int\cos x\,dx=\sin x+C, \qquad \int\sin x\,dx=-\cos 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 $\int(x^3+2\sin x)\,dx$ .

Your work stays on this device. No account or AI grader is used.

Show hint

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

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Basic Antiderivative Rules

Explanation

Integration rules are derivative rules read backward

For

n\ne-1

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

Other central formulas include

\int e^x\,dx=e^x+C, \qquad \int\frac1x\,dx=\ln|x|+C,

\int\cos x\,dx=\sin x+C, \qquad \int\sin x\,dx=-\cos x+C,

\int\sec^2x\,dx=\tan x+C, \qquad \int\frac1{1+x^2}\,dx=\arctan x+C.

Decision

Normalize before integrating

Interactive checku3a-basic-rules-01

Evaluate $\int(x^3+2\sin x)\,dx$ .

Your work stays on this device. No account or AI grader is used.

Show hint

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

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Basic Antiderivative Rules

Related resources

Original instruction with traceable references.

Basic Antiderivative Rules

Related resources

Original instruction with traceable references.