Integration Formulas with Conditions, Examples and PDF

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Basic antiderivatives

Reverse familiar derivative rules for powers, exponentials, reciprocals, sine, and cosine.

$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$: Power antiderivativen is constant with respect to x. n is not negative one. Use an interval where the real-valued power is defined.
Use this rule to integrate a power of x other than x to the negative one.
$\int e^x\,dx=e^x+C$: Natural exponential antiderivativeC is an arbitrary constant of integration.
Use this rule to integrate the natural exponential function.
$\int \frac{1}{x}\,dx=\ln|x|+C$: Reciprocal antiderivativex is not zero. Choose an interval entirely on one side of zero. C is an arbitrary constant of integration.
Use this rule for the reciprocal function on an interval that does not cross zero.
$\int \sin x\,dx=-\cos x+C$: Sine antiderivativeAngles are measured in radians. C is an arbitrary constant of integration.
Use this rule for sine of the integration variable; use substitution when the angle is a nontrivial inner function.
$\int \cos x\,dx=\sin x+C$: Cosine antiderivativeAngles are measured in radians. C is an arbitrary constant of integration.
Use this rule for cosine of the integration variable; use substitution when the angle is a nontrivial inner function.

Linearity and methods

Separate sums and constants, or transform the integral when a direct rule does not fit.

$\int[af(x)+bg(x)]\,dx=a\int f(x)\,dx+b\int g(x)\,dx$: Linearity of integrationa and b are constant with respect to x. The displayed antiderivatives exist on a common interval.
Use linearity to separate a sum and move constant factors outside the integral.
$\int f(g(x))g'(x)\,dx=\int f(u)\,du,\quad u=g(x)$: Substitution ruleg is differentiable on the interval of substitution. The substitution preserves the relevant interval and domain.
Use substitution when the integrand contains a composition together with the derivative of its inner function, up to a constant factor.
$\int u\,dv=uv-\int v\,du$: Integration by partsu and v are differentiable on the interval of interest. The displayed integrals exist on that interval.
Use this identity when differentiating one factor simplifies it and the other factor can be integrated.

Definite integrals

Use a verified antiderivative for endpoint evaluation or an average value on a closed interval.

$\int_a^b f(x)\,dx=F(b)-F(a)$: Definite integral evaluationf is continuous on the closed interval from a to b. F is an antiderivative of f on that interval.
Use this result to evaluate a definite integral from a verified antiderivative.
$f_{\mathrm{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx$: Average value of a functionf is integrable on the closed interval from a to b. b is greater than a.
Use this formula to find the continuous average of a function over a finite interval.

Notation

Symbols used on this sheet

Symbol	Meaning	Usage note
$\int f(x)\,dx$	Indefinite integral	Represents a family of antiderivatives and therefore includes C.
$\int_a^b f(x)\,dx$	Definite integral	Represents signed accumulation from a to b.
$C$	Constant of integration	Any real constant produces another member of the antiderivative family.
$F'(x)=f(x)$	Antiderivative relationship	Differentiating F independently checks an indefinite result.
$u,v$	Functions used in integration by parts	Choose them so the remaining integral is simpler and all required derivatives exist.
$a,b$	Interval endpoints	Reversing the endpoints changes the sign of a definite integral.

Use the reference

Short applications

These examples show when to choose a formula; detailed instruction belongs in the linked guide or calculator.

Recognize an inner derivative

The factor 2x is the derivative of x squared, so substitution converts the integral to a basic sine antiderivative.

\int2x\cos(x^2)\,dx

\sin(x^2)+C

Condition check: Differentiating sine of x squared by the chain rule returns 2x cosine of x squared.

Evaluate with the Fundamental Theorem

The polynomial is continuous and has x cubed as an antiderivative.

\int_0^2 3x^2\,dx

[x^3]_0^2=8

Condition check: A Riemann-sum or numerical quadrature check approaches the same positive accumulated value.

Use conditions, not memory alone

Common formula confusions

Using the power formula at n equals negative one

Dividing by n plus one when that value is zero.

Fix: Use the logarithmic antiderivative for one over x on an interval that does not cross zero.

Omitting the integration constant

Reporting one antiderivative as the entire indefinite-integral family.

Fix: Add C after every indefinite integral and omit it only when evaluating a definite integral.

Losing bounds during a method

Switching variables or applying integration by parts without carrying endpoint information consistently.

Fix: Either transform every bound with the variable or return to the original variable before evaluating.

Calling signed accumulation total area

Assuming negative graph regions contribute positively to a definite integral.

Fix: Split at sign changes and integrate absolute value when the task asks for geometric area.

Common questions

Integration formulas FAQ

Which integration formulas are included?

The focused sheet includes power, natural exponential, reciprocal, integration-by-parts, and definite-integral evaluation formulas with their conditions.

When should an integral include plus C?

Every indefinite integral needs an arbitrary constant because differentiation removes constants. A definite integral evaluates endpoint change and does not include plus C.

How can I check an antiderivative?

Differentiate it on the stated domain. The derivative must reproduce the original integrand exactly, including inner-derivative scale factors.

Does the integration PDF match this page?

Yes. The PDF and HTML are generated from the same structured formula source, including names, notation, and usage conditions.

Sources and curriculum alignment

This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.

5.3 The Fundamental Theorem of Calculus, OpenStax Calculus Volume 1
3.1 Integration by Parts, OpenStax Calculus Volume 2

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