Table shows indefinite integrals (antiderivative) of selected functions.

Integrals - formulas#

Function $f(x)$	Integral $\int{f(x)dx}$	Notice
Show source $a$	Show source $ax + C$	Show source
Show source $x$	Show source $\dfrac{1}{2} x^2 + C$	Show source
Show source $x^n$	Show source $\dfrac{1}{n + 1} x^{n+1} + C$	Show source $n \neq -1$
Show source $\dfrac{1}{x}$	Show source $ln\left \| x \right \| + C$	Show source
Show source $a^x$	Show source $\frac{1}{ln \: a}a^x + C$	Show source
Show source $ln \: x$	Show source $(x-1) \: ln \: x + C$	Show source
Show source $log_a x$	Show source $\dfrac{x}{ln \: a}(ln \: x - 1) + C$	Show source
Show source $e^x$	Show source $e^x + C$	Show source
Show source $\sqrt{x}$	Show source $\dfrac{2}{3} \sqrt{x^3} + C$	Show source
Show source $\dfrac{1}{\sqrt{x}}$	Show source $2 \sqrt{x} + C$	Show source
Show source $\dfrac{1}{ax +b}$	Show source $\dfrac{1}{a} ln \left \|ax +b \right \|+ C$	Show source $a \neq 0$
Show source $sin \: x$	Show source $- cos \: x + C$	Show source
Show source $cos \: x$	Show source $sin \: x + C$	Show source
Show source $tg \: x$	Show source $-ln \left\| cos \: x \right\|+ C$	Show source
Show source $ctg \: x$	Show source $ln \left\| sin \: x \right\|+ C$	Show source
Show source $\dfrac{1}{cos^2 x}$	Show source $tg \: x + C$	Show source $cos \: x \neq 0$
Show source $\dfrac{1}{sin^2 x}$	Show source $-ctg \: x + C$	Show source $sin \: x \neq 0$
Show source $\dfrac{1}{x^2 +a^2}$	Show source $\dfrac{1}{a} arc \: tg \dfrac{x}{a} + C$	Show source $a \neq 0$
Show source $\dfrac{1}{\sqrt{a^2 - x^2}}$	Show source $arc \: sin \dfrac{x}{a} + C$	Show source $a \neq 0$
Show source $\dfrac{1}{\sqrt{x^2 - a^2}}$	Show source $ln \left\| x+ \sqrt{x^2 - a^2} \right \| + C$	Show source
Show source $(ax + b)^n$	Show source $\dfrac{1}{a(n+1)} (ax + b)^{n+1} + C$	Show source $n \neq -1$
Show source $\dfrac{1}{a^2 - x^2}$	Show source $\dfrac{1}{2a}ln \left\|\dfrac{a+x}{a-x} \right\| +C$	Show source $a>0, \: \left\|x \right\| \neq a$

Some facts#

The indefinite integral is a function.
Integration is a process opposite to the derivation (differentiation). The integral of f(x) is s(x), if it's derivative reproduces this function:

$\int f(x) = s(x),\text{ if }\dfrac{ds}{dx} = f(x)$
Function s(x) is sometimes called antiderivative of f(x) or intrinsic function.
If f(x) is integral of some function, then each function in below form is also it's integral:

$f(x) + C$
where C is arbitrary constant. It's so-called integration constant.
This property results from the fact that derivative from the constant (C) function is equal to 0 at each point.

ⓘ Example: The integral of polynomial $3x ^ 2 + 2x + 5$ is:
$\int (3x^2 + 2x + 5) dx = x^3 + x^2 + 5x + C$
because when we calculate it's derivative, then we'll get back this polynomial:
$\dfrac{d}{dx} x^3 + x^2 + 5x + C = 3x^2 + 2x + 5$
In contrast to derivatives there are no ready-made formulas that can calculate the integral of any function in routine way. In general, integration requires more sophisticated methods adapted to the specific problem.
Not every function has its intrinsic function. In other words, there are functions whose integral does not exist.
Many practical problems e.g. in the field of natural or technical sciences, lead to the need of calculating one or more integrals at some point.
The equation containing integral from the unknown function is called integral equation.

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integral_formulas · indefinite_integrals · calculus · antiderivative · intrinsic_functions

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