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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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Reference mode: click any card to see the full formula and details.

Polynomial
kk
Constant rule
↓
kx+Ckx + C
Polynomial
xx
Identity
↓
x22+C\dfrac{x^2}{2} + C
Polynomial
xnx^n
Power rule
↓
xn+1n+1+C\dfrac{x^{n+1}}{n+1} + C
Polynomial
x\sqrt{x}
Square root
↓
23 x3/2+C\dfrac{2}{3}\,x^{3/2} + C
Polynomial
1x2\dfrac{1}{x^2}
Reciprocal square
↓
−1x+C-\dfrac{1}{x} + C
Exponential
exe^x
Natural exponential
↓
ex+Ce^x + C
Exponential
ekxe^{kx}
Scaled exponential
↓
ekxk+C\dfrac{e^{kx}}{k} + C
Exponential
axa^x
General exponential
↓
axln⁡a+C\dfrac{a^x}{\ln a} + C
Logarithmic
1x\dfrac{1}{x}
Reciprocal
↓
ln⁡∣x∣+C\ln|x| + C
Logarithmic
ln⁡(x)\ln(x)
Natural logarithm
↓
xln⁡(x)−x+Cx\ln(x) - x + C
Trigonometry
sin⁡(x)\sin(x)
Sine
↓
−cos⁡(x)+C-\cos(x) + C
Trigonometry
cos⁡(x)\cos(x)
Cosine
↓
sin⁡(x)+C\sin(x) + C
Trigonometry
tan⁡(x)\tan(x)
Tangent
↓
ln⁡∣sec⁡(x)∣+C\ln|\sec(x)| + C
Trigonometry
cot⁡(x)\cot(x)
Cotangent
↓
ln⁡∣sin⁡(x)∣+C\ln|\sin(x)| + C
Trigonometry
sec⁡(x)\sec(x)
Secant
↓
ln⁡∣sec⁡(x)+tan⁡(x)∣+C\ln|\sec(x) + \tan(x)| + C
Trigonometry
csc⁡(x)\csc(x)
Cosecant
↓
−ln⁡∣csc⁡(x)+cot⁡(x)∣+C-\ln|\csc(x) + \cot(x)| + C
Trigonometry
sec⁡2(x)\sec^2(x)
Secant squared
↓
tan⁡(x)+C\tan(x) + C
Trigonometry
csc⁡2(x)\csc^2(x)
Cosecant squared
↓
−cot⁡(x)+C-\cot(x) + C
Trigonometry
sec⁡(x)tan⁡(x)\sec(x)\tan(x)
Secant · Tangent
↓
sec⁡(x)+C\sec(x) + C
Trigonometry
csc⁡(x)cot⁡(x)\csc(x)\cot(x)
Cosecant · Cotangent
↓
−csc⁡(x)+C-\csc(x) + C
Inverse trigonometry
11−x2\dfrac{1}{\sqrt{1 - x^2}}
Arcsine integrand
↓
arcsin⁡(x)+C\arcsin(x) + C
Inverse trigonometry
11+x2\dfrac{1}{1 + x^2}
Arctangent integrand
↓
arctan⁡(x)+C\arctan(x) + C
Inverse trigonometry
1xx2−1\dfrac{1}{x\sqrt{x^2 - 1}}
Arcsecant integrand
↓
arcsec⁡∣x∣+C\operatorname{arcsec}|x| + C
Hyperbolic
sinh⁡(x)\sinh(x)
Hyperbolic sine
↓
cosh⁡(x)+C\cosh(x) + C
Hyperbolic
cosh⁡(x)\cosh(x)
Hyperbolic cosine
↓
sinh⁡(x)+C\sinh(x) + C
Hyperbolic
tanh⁡(x)\tanh(x)
Hyperbolic tangent
↓
ln⁡(cosh⁡(x))+C\ln(\cosh(x)) + C
Hyperbolic
sech⁡2(x)\operatorname{sech}^2(x)
Hyperbolic sech²
↓
tanh⁡(x)+C\tanh(x) + C
Hyperbolic
csch⁡2(x)\operatorname{csch}^2(x)
Hyperbolic csch²
↓
−coth⁡(x)+C-\coth(x) + C
Hyperbolic
sech⁡(x)tanh⁡(x)\operatorname{sech}(x)\tanh(x)
sech · tanh
↓
−sech⁡(x)+C-\operatorname{sech}(x) + C
Hyperbolic
csch⁡(x)coth⁡(x)\operatorname{csch}(x)\coth(x)
csch · coth
↓
−csch⁡(x)+C-\operatorname{csch}(x) + C

Families of integrals

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

x^n

Polynomial

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

5 matchesClick to highlight
e^x

Exponential

Natural and general exponential integrands.

3 matchesClick to highlight
ln

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.

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

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.

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

Substitution

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

∫ ⁣f(g(x)) g(ˊx) dx=∫ ⁣f(u) du\displaystyle \int\! f(g(x))\, g\'(x)\, dx = \int\! f(u)\, du
uv

Integration by parts

Reverses the product rule. Choose uu so it simplifies under differentiation, and dvdv so you can integrate it.

∫ ⁣u dv=uv−∫ ⁣v du\displaystyle \int\! u\, dv = uv - \int\! v\, du
↔

Inverse of differentiation

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

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