Table of indefinite integrals of selected functions
Table shows indefinite integrals (antiderivative) of selected functions.

# Integrals - formulas

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

# 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:
$\int f(x) = s(x),\text{ if }\frac{ds}{dx} = f(x)$
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:
$\frac{d}{dx} x^3 + x^2 + 5x + C = 3x^2 + 2x + 5$
• 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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