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
Show sourceaaShow sourceax+Cax + CShow source
Show sourcexxShow source12x2+C\dfrac{1}{2} x^2 + CShow source
Show sourcexnx^nShow source1n+1xn+1+C\dfrac{1}{n + 1} x^{n+1} + CShow sourcen1n \neq -1
Show source1x\dfrac{1}{x}Show sourcelnx+Cln\left | x \right | + CShow source
Show sourceaxa^xShow source1lnaax+C\frac{1}{ln \: a}a^x + CShow source
Show sourcelnxln \: xShow source(x1)lnx+C(x-1) \: ln \: x + CShow source
Show sourcelogaxlog_a xShow sourcexlna(lnx1)+C\dfrac{x}{ln \: a}(ln \: x - 1) + CShow source
Show sourceexe^xShow sourceex+Ce^x + CShow source
Show sourcex\sqrt{x}Show source23x3+C\dfrac{2}{3} \sqrt{x^3} + CShow source
Show source1x\dfrac{1}{\sqrt{x}}Show source2x+C2 \sqrt{x} + CShow source
Show source1ax+b\dfrac{1}{ax +b}Show source1alnax+b+C\dfrac{1}{a} ln \left |ax +b \right |+ CShow sourcea0a \neq 0
Show sourcesinxsin \: xShow sourcecosx+C- cos \: x + CShow source
Show sourcecosxcos \: xShow sourcesinx+Csin \: x + CShow source
Show sourcetgxtg \: xShow sourcelncosx+C-ln \left| cos \: x \right|+ CShow source
Show sourcectgxctg \: xShow sourcelnsinx+Cln \left| sin \: x \right|+ CShow source
Show source1cos2x\dfrac{1}{cos^2 x}Show sourcetgx+Ctg \: x + CShow sourcecosx0cos \: x \neq 0
Show source1sin2x\dfrac{1}{sin^2 x}Show sourcectgx+C-ctg \: x + CShow sourcesinx0sin \: x \neq 0
Show source1x2+a2\dfrac{1}{x^2 +a^2}Show source1aarctgxa+C\dfrac{1}{a} arc \: tg \dfrac{x}{a} + CShow sourcea0a \neq 0
Show source1a2x2\dfrac{1}{\sqrt{a^2 - x^2}}Show sourcearcsinxa+Carc \: sin \dfrac{x}{a} + CShow sourcea0a \neq 0
Show source1x2a2\dfrac{1}{\sqrt{x^2 - a^2}}Show sourcelnx+x2a2+Cln \left| x+ \sqrt{x^2 - a^2} \right | + CShow source
Show source(ax+b)n(ax + b)^nShow source1a(n+1)(ax+b)n+1+C\dfrac{1}{a(n+1)} (ax + b)^{n+1} + CShow sourcen1n \neq -1
Show source1a2x2\dfrac{1}{a^2 - x^2}Show source12alna+xax+C\dfrac{1}{2a}ln \left|\dfrac{a+x}{a-x} \right| +CShow sourcea>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 }\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)+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\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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