Formulas and Tables

Calculus

Properties of Indefinite Integrals

Functions: \(f\), \(g\), \(u\), \(v\), \(F\)
Independent variables: \(x\), \(t\)

Constant real numbers: \(C\), \(a\), \(b\), \(k\)

  1. An antiderivative of a function \(y = f\left( x \right)\) defined on some interval \(\left( {a,b} \right)\) is called any function \(F\left( x \right)\) whose derivative at any point of this interval is equal to \(f\left( x \right)\):
    \(F’\left( x \right) = f\left( x \right)\).
    If \(F\left( x \right)\) is an antiderivative of \(f\left( x \right)\), then the function of the form \(F\left( x \right) + C\), where \(C\) is an arbitrary constant, is also an antiderivative of \(f\left( x \right)\).
  2. The indefinite integral of a function \(f\left( x \right)\) is the collection of all antiderivatives for this function:
    \(\int {f\left( x \right)dx} = F\left( x \right) + C,\) if \(F’\left( x \right) = f\left( x \right).\)
  3. The derivative of the indefinite integral is equal to the integrand:
    \({\left( {\int {f\left( x \right)dx} } \right)^\prime } = f\left( x \right).\)
  4. The indefinite integral of the sum of two functions is equal to the sum of the integrals:
    \(\int {\left[ {f\left( x \right) + g\left( x \right)} \right]dx} =\) \(\int {f\left( x \right)dx} + \int {g\left( x \right)dx} .\)
  5. The indefinite integral of the difference of two functions is equal to the difference of the integrals:
    \(\int {\left[ {f\left( x \right) – g\left( x \right)} \right]dx} =\) \(\int {f\left( x \right)dx} – \int {g\left( x \right)dx} .\)
  6. A constant factor can be moved across the integral sign:
    \(\int {kf\left( x \right)dx} = k\int {f\left( x \right)dx} .\)
  7. \(\int {f\left( {ax} \right)dx} =\) \({\large\frac{1}{a}\normalsize} F\left( {ax} \right) + C\)
  8. \(\int {f\left( {ax + b} \right)dx} =\) \({\large\frac{1}{a}\normalsize} F\left( {ax + b} \right) + C\)
  9. \(\int {f\left( x \right)f’\left( x \right)dx} =\) \({\large\frac{1}{2}\normalsize} {f^2}\left( x \right) + C\)
  10. \(\int {{\large\frac{{f’\left( x \right)}}{{f\left( x \right)}}\normalsize} dx} =\) \(\ln \left| {f\left( x \right)} \right| + C\)
  11. Integration by substitution
    \(\int {f\left( x \right)dx} = \int {f\left( {u\left( t \right)} \right)u’\left( t \right)dt} ,\) if \(x = u\left( t \right).\)
  12. Integration by parts
    \(\int {udv} = uv – \int {vdu} ,\)
    where \(u\left( x \right)\), \(v\left( x \right)\) are differentiable functions.