Indefinite Integral Formulas

Posted In: Mathematics

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

Functions: $f,g,u,v$
Independent variables: $x,t,ξ$
Indefinite integral of a function: $∫f(x)dx,∫g(x)dx$
Derivative of a function: $y'(x), f'(x),F'(x)$
Real constants: a, b, c, d, k
Natural numbers: m, n, i, j

  1. $∫f(x)\ dx=F(x)+c,$ if $F'(x)=f(x)$
  2. $(∫f(x)\ dx)’=f(x)$
  3. $∫kf(x)\ dx=k∫f(x)\ dx$
  4. $∫[f(x)+g(x)]\ dx=∫f(x)\ dx+∫g(x)\ dx$
  5. $∫[f(x)-g(x)]\ dx=∫f(x)\ dx-∫g(x)\ dx$
  6. $∫f(ax)\ dx=1/aF(ax)+c$
  7. $∫f(ax+b)\ dx=1/aF(ax+b)+c$
  8. $∫f(x)f'(x)\ dx=1/2f^2(x)+c$
  9. $∫{f'(x)}/{f(x)}\ dx=ln\ |f(x)|+c$
  10. Method of substitution
    $∫f(x)\ dx=∫f(u(t))u'(t)\ dt,$ if $x=u(t)$
  11. Integratiomn by parts
    $∫udv=uv-∫v\ du$
    where $u(x),v(x)$ are differential functions.

Post Tags: Integration

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