Calculus I - Antidifferentiation / Indefinite Integral Rules

7 flashcards · Mathematics · by luckydog789 (harrison holbrook)

1. Indefinite integral notation

   F(x)=∫f(x)dxF(x)=\int f(x)dxF(x)=∫f(x)dx F(x)=F(x)=F(x)= antiderivative of f(x)f(x)f(x) ∫=\int =∫= integral sign f(x)f(x)f(x) = integrand dxdxdx = variable being operated on These are not just notational choices, but rather reflect intuitive ideas of calculus (think Riemann Sums)

2. Antiderivative

   A function FFF is an antiderivative of fff on an interval III if F′(x)=f(x)F'(x)=f(x)F′(x)=f(x) for all xxx in III

3. Constant of Integration

   A constant term added to an antiderivative to generalize it to represent all possible antiderivatives of a function, as the constant term does not survive differentiation and therefore does not change the original function being antidifferentiated. This term is represented as CCC

4. Family of Antiderivatives

   Let F(x)F(x)F(x) be any antiderivative of f(x)f(x)f(x) on an interval III . Then all antiderivatives of fff on III have the form F(x)+CF(x)+CF(x)+C where CCC is the constant of integration.

5. Power rule of antiderivatives

   ∫xpdx=xp+1p+1+C\int x^pdx=\frac{x^{p+1}}{p+1}+C∫xpdx=p+1xp+1​+C , if p≠0p \neq 0p=0

6. Constant multiple rule of antiderivatives

   ∫cf(x)dx=c∫f(x)dx\int cf(x)dx=c\int f(x)dx ∫cf(x)dx=c∫f(x)dx

7. Sun rule of antiderivatives

   ∫(f(x)±g(x))dx=∫f(x)dx±∫g(x)dx\int(f(x) \pm g(x))dx=\int f(x)dx \pm \int g(x)dx∫(f(x)±g(x))dx=∫f(x)dx±∫g(x)dx