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Calculus reference guide

Common antiderivatives and integration rules

The formulas and properties calculus students are most often expected to recognize, recall, and apply.

Every indefinite integral needs + CAngles are in radiansCheck results by differentiating
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What should I memorize first?

Start with the power, reciprocal, exponential, sine, and cosine antiderivatives. Then learn the remaining trig pairs and the substitution patterns. Definite-integral properties and integration by parts matter once those foundations are automatic.

01

Core antiderivative rules

Every indefinite integral represents a family of functions, so include the constant of integration.

Constant rule

∫c dx=cx+C\int c\,dx=cx+C

Power rule

Valid for n ≠ −1.

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

Constant multiple rule

∫cf(x) dx=c∫f(x) dx\int c f(x)\,dx=c\int f(x)\,dx

Sum and difference rule

∫[f(x)±g(x)]dx=∫f(x) dx±∫g(x) dx\int\left[f(x)\pm g(x)\right]dx=\int f(x)\,dx\pm\int g(x)\,dx
02

Exponential and logarithmic forms

The logarithmic form is the missing n = −1 case from the power rule.

Natural exponential

∫ex dx=ex+C\int e^x\,dx=e^x+C

Exponential with base a

Assume a > 0 and a ≠ 1.

∫ax dx=axln⁡a+C\int a^x\,dx=\frac{a^x}{\ln a}+C

Reciprocal

Valid on intervals that do not cross x = 0.

∫1x dx=ln⁡∣x∣+C\int\frac{1}{x}\,dx=\ln|x|+C
03

Trigonometric antiderivatives

These six pairs reverse the standard trigonometric derivative formulas.

Cosine

∫cos⁡x dx=sin⁡x+C\int\cos x\,dx=\sin x+C

Sine

∫sin⁡x dx=−cos⁡x+C\int\sin x\,dx=-\cos x+C

Secant squared

∫sec⁡2x dx=tan⁡x+C\int\sec^2 x\,dx=\tan x+C

Cosecant squared

∫csc⁡2x dx=−cot⁡x+C\int\csc^2 x\,dx=-\cot x+C

Secant times tangent

∫sec⁡xtan⁡x dx=sec⁡x+C\int\sec x\tan x\,dx=\sec x+C

Cosecant times cotangent

∫csc⁡xcot⁡x dx=−csc⁡x+C\int\csc x\cot x\,dx=-\csc x+C
04

Inverse trigonometric forms

Recognize the entire denominator pattern before choosing an inverse trigonometric result.

Inverse sine form

Assume a > 0.

∫1a2−x2 dx=arcsin⁡(xa)+C\int\frac{1}{\sqrt{a^2-x^2}}\,dx=\arcsin\left(\frac{x}{a}\right)+C

Inverse tangent form

Assume a > 0.

∫1a2+x2 dx=1aarctan⁡(xa)+C\int\frac{1}{a^2+x^2}\,dx=\frac{1}{a}\arctan\left(\frac{x}{a}\right)+C

Inverse secant form

Assume a > 0 and |x| > a.

∫1xx2−a2 dx=1aarcsec⁡(∣x∣a)+C\int\frac{1}{x\sqrt{x^2-a^2}}\,dx=\frac{1}{a}\operatorname{arcsec}\left(\frac{|x|}{a}\right)+C
05

Common substitution patterns

Let u be a differentiable function of x. Each pattern includes the inside derivative u′.

Power of a function

Valid for n ≠ −1.

∫unu′ dx=un+1n+1+C\int u^n u'\,dx=\frac{u^{n+1}}{n+1}+C

Logarithmic pattern

∫u′u dx=ln⁡∣u∣+C\int\frac{u'}{u}\,dx=\ln|u|+C

Natural exponential pattern

∫euu′ dx=eu+C\int e^u u'\,dx=e^u+C

Sine pattern

∫sin⁡(u)u′ dx=−cos⁡(u)+C\int\sin(u)u'\,dx=-\cos(u)+C

Cosine pattern

∫cos⁡(u)u′ dx=sin⁡(u)+C\int\cos(u)u'\,dx=\sin(u)+C

Inverse tangent pattern

∫u′1+u2 dx=arctan⁡(u)+C\int\frac{u'}{1+u^2}\,dx=\arctan(u)+C
06

Definite-integral properties

These rules reorganize bounds and split or combine accumulated change.

Same bounds

∫aaf(x) dx=0\int_a^a f(x)\,dx=0

Reverse the bounds

∫abf(x) dx=−∫baf(x) dx\int_a^b f(x)\,dx=-\int_b^a f(x)\,dx

Add adjacent intervals

∫abf(x) dx+∫bcf(x) dx=∫acf(x) dx\int_a^b f(x)\,dx+\int_b^c f(x)\,dx=\int_a^c f(x)\,dx

Linearity

∫ab[cf(x)+g(x)]dx=c∫abf(x)dx+∫abg(x)dx\int_a^b[cf(x)+g(x)]dx=c\int_a^b f(x)dx+\int_a^b g(x)dx

Fundamental Theorem

∫abF′(x) dx=F(b)−F(a)\int_a^b F'(x)\,dx=F(b)-F(a)
07

Two essential technique formulas

Substitution reverses the chain rule; integration by parts reverses the product rule.

Substitution

Set u = g(x).

∫f(g(x))g′(x) dx=∫f(u) du\int f(g(x))g'(x)\,dx=\int f(u)\,du

Integration by parts

∫u dv=uv−∫v du\int u\,dv=uv-\int v\,du
Best error check

Differentiate your answer.

An antiderivative is correct when its derivative returns the original integrand. This quick check catches missing chain-rule factors, sign errors, and incorrect exponents.