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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.