What are common integrals?

Common integrals are the indefinite integral identities used most often in calculus: integrals of polynomials, exponentials, logarithms, the six trigonometric functions, key inverse trigonometric integrands, and hyperbolic functions. Every indefinite integral is determined only up to an additive constant of integration, written as plus C.

What is the integral of sin x and cos x?

The integral of sin(x) is negative cos(x) plus C. The integral of cos(x) is sin(x) plus C. The sign flip on sine reflects the derivative relationship: the derivative of cos(x) is negative sin(x), so reversing the differentiation introduces the negative sign.

Why does 1 over x integrate to ln of x?

The integral of 1/x is ln|x| plus C. This is the one exception to the power rule, which fails when the exponent equals negative one because dividing by n plus one would divide by zero. The absolute value extends the formula to negative x, since ln is only defined for positive arguments.

What is the constant of integration?

An indefinite integral is determined only up to an additive constant because the derivative of any constant is zero. If F(x) is one antiderivative of f(x), then F(x) plus C is also an antiderivative for every real constant C. This is why every entry in an indefinite integrals table ends with plus C.

How do I memorize common integrals?

Group the entries by category: polynomial via the power rule, exponential, logarithmic, trigonometric, inverse trigonometric, and hyperbolic. Most integrals are derivatives read backwards, so memorizing the derivatives table cuts the work in half. Drill with the drag-and-drop puzzle on this page to lock the pairings in.

Common Integrals

Reference table of common indefinite integrals. Try puzzle mode to drill, or read the full integration explanation →

Indefinite integral tool

Search by integrand, by antiderivative, by name — or pick a family.

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Name	Integrand		Antiderivative	Family
Constant rule	$k$	=	$kx + C$	Polynomial
Identity	$x$	=	$\dfrac{x^2}{2} + C$	Polynomial
Power rule	$x^n$	=	$\dfrac{x^{n+1}}{n+1} + C$	Polynomial
Square root	$\sqrt{x}$	=	$\dfrac{2}{3}\,x^{3/2} + C$	Polynomial
Reciprocal square	$\dfrac{1}{x^2}$	=	$-\dfrac{1}{x} + C$	Polynomial
Natural exponential	$e^x$	=	$e^x + C$	Exponential
Scaled exponential	$e^{kx}$	=	$\dfrac{e^{kx}}{k} + C$	Exponential
General exponential	$a^x$	=	$\dfrac{a^x}{\ln a} + C$	Exponential
Reciprocal	$\dfrac{1}{x}$	=	$\ln\|x\| + C$	Logarithmic
Natural logarithm	$\ln(x)$	=	$x\ln(x) - x + C$	Logarithmic
Sine	$\sin(x)$	=	$-\cos(x) + C$	Trigonometry
Cosine	$\cos(x)$	=	$\sin(x) + C$	Trigonometry
Tangent	$\tan(x)$	=	$\ln\|\sec(x)\| + C$	Trigonometry
Cotangent	$\cot(x)$	=	$\ln\|\sin(x)\| + C$	Trigonometry
Secant	$\sec(x)$	=	$\ln\|\sec(x) + \tan(x)\| + C$	Trigonometry
Cosecant	$\csc(x)$	=	$-\ln\|\csc(x) + \cot(x)\| + C$	Trigonometry
Secant squared	$\sec^2(x)$	=	$\tan(x) + C$	Trigonometry
Cosecant squared	$\csc^2(x)$	=	$-\cot(x) + C$	Trigonometry
Secant · Tangent	$\sec(x)\tan(x)$	=	$\sec(x) + C$	Trigonometry
Cosecant · Cotangent	$\csc(x)\cot(x)$	=	$-\csc(x) + C$	Trigonometry
Arcsine integrand	$\dfrac{1}{\sqrt{1 - x^2}}$	=	$\arcsin(x) + C$	Inverse trigonometry
Arctangent integrand	$\dfrac{1}{1 + x^2}$	=	$\arctan(x) + C$	Inverse trigonometry
Arcsecant integrand	$\dfrac{1}{x\sqrt{x^2 - 1}}$	=	$\operatorname{arcsec}\|x\| + C$	Inverse trigonometry
Hyperbolic sine	$\sinh(x)$	=	$\cosh(x) + C$	Hyperbolic
Hyperbolic cosine	$\cosh(x)$	=	$\sinh(x) + C$	Hyperbolic
Hyperbolic tangent	$\tanh(x)$	=	$\ln(\cosh(x)) + C$	Hyperbolic
Hyperbolic sech²	$\operatorname{sech}^2(x)$	=	$\tanh(x) + C$	Hyperbolic
Hyperbolic csch²	$\operatorname{csch}^2(x)$	=	$-\coth(x) + C$	Hyperbolic
sech · tanh	$\operatorname{sech}(x)\tanh(x)$	=	$-\operatorname{sech}(x) + C$	Hyperbolic
csch · coth	$\operatorname{csch}(x)\coth(x)$	=	$-\operatorname{csch}(x) + C$	Hyperbolic

Families of integrals

Click a family to highlight its entries in the table above.

x^n

Polynomial

Powers of $x$ , roots, and reciprocals — the power rule, reversed.

5 matchesClick to highlight

e^x

Exponential

Natural and general exponential integrands.

3 matchesClick to highlight

Logarithmic

Reciprocal and logarithmic integrands.

2 matchesClick to highlight

sin

Trigonometry

All six basic trigonometric functions and their squared / product forms.

10 matchesClick to highlight

arc

Inverse trigonometry

Integrands whose antiderivative is an inverse trigonometric function.

3 matchesClick to highlight

sinh

Hyperbolic

Hyperbolic sine, cosine, and their relatives.

7 matchesClick to highlight

Properties of indefinite integration

Rules that combine and transform integrals.

Linearity

The integral of a sum is the sum of integrals, and a constant multiplier passes through the integral sign.

\displaystyle \int \!\bigl(af(x) + bg(x)\bigr)\, dx = a\!\int\! f(x)\, dx + b\!\int\! g(x)\, dx

Constant of integration

An indefinite integral is determined only up to an additive constant. Any two antiderivatives of the same function differ by a constant.

\displaystyle \int\! f(x)\, dx = F(x) + C

Substitution

When the integrand contains a function and its derivative, substitute $u = g(x)$ so that $du = g\'(x)\, dx$ . Reverses the chain rule.

\displaystyle \int\! f(g(x))\, g\'(x)\, dx = \int\! f(u)\, du

Integration by parts

Reverses the product rule. Choose $u$ so it simplifies under differentiation, and $dv$ so you can integrate it.

\displaystyle \int\! u\, dv = uv - \int\! v\, du

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Inverse of differentiation

Integration and differentiation are inverse operations. Every entry in the derivatives table is also an integral, read backwards.

\displaystyle \dfrac{d}{dx}\!\left[\int\! f(x)\, dx\right] = f(x)