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

Integrals - formulas

Function f(x)f(x)Integral f(x)dx\int{f(x)dx}Notice
aaax+Cax + C
xx12x2+C\dfrac{1}{2} x^2 + C
xnx^n1n+1xn+1+C\dfrac{1}{n + 1} x^{n+1} + Cn1n \neq -1
1x\dfrac{1}{x}lnx+Cln\left | x \right | + C
axa^x1lnaax+C\frac{1}{ln \: a}a^x + C
lnxln \: x(x1)lnx+C(x-1) \: ln \: x + C
logaxlog_a xxlna(lnx1)+C\dfrac{x}{ln \: a}(ln \: x - 1) + C
exe^xex+Ce^x + C
x\sqrt{x}23x3+C\dfrac{2}{3} \sqrt{x^3} + C
1x\dfrac{1}{\sqrt{x}}2x+C2 \sqrt{x} + C
1ax+b\dfrac{1}{ax +b}1alnax+b+C\dfrac{1}{a} ln \left |ax +b \right |+ Ca0a \neq 0
sinxsin \: xcosx+C- cos \: x + C
cosxcos \: xsinx+Csin \: x + C
tgxtg \: xlncosx+C-ln \left| cos \: x \right|+ C
ctgxctg \: xlnsinx+Cln \left| sin \: x \right|+ C
1cos2x\dfrac{1}{cos^2 x}tgx+Ctg \: x + Ccosx0cos \: x \neq 0
1sin2x\dfrac{1}{sin^2 x}ctgx+C-ctg \: x + Csinx0sin \: x \neq 0
1x2+a2\dfrac{1}{x^2 +a^2}1aarctgxa+C\dfrac{1}{a} arc \: tg \dfrac{x}{a} + Ca0a \neq 0
1a2x2\dfrac{1}{\sqrt{a^2 - x^2}}arcsinxa+Carc \: sin \dfrac{x}{a} + Ca0a \neq 0
1x2a2\dfrac{1}{\sqrt{x^2 - a^2}}lnx+x2a2+Cln \left| x+ \sqrt{x^2 - a^2} \right | + C
(ax+b)n(ax + b)^n1a(n+1)(ax+b)n+1+C\dfrac{1}{a(n+1)} (ax + b)^{n+1} + Cn1n \neq -1
1a2x2\dfrac{1}{a^2 - x^2}12alna+xax+C\dfrac{1}{2a}ln \left|\dfrac{a+x}{a-x} \right| +Ca>0,xaa>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:
    f(x)=s(x), if dsdx=f(x)\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)+Cf(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 3x2+2x+53x ^ 2 + 2x + 5 is:
    (3x2+2x+5)dx=x3+x2+5x+C\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:
    ddxx3+x2+5x+C=3x2+2x+5\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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