Formulas

Antiderivatives

In the following list, one antiderivative is given for each function. To find all antiderivatives for that function, add an arbitrary constant `C`.

Function	Antiderivative
`x^n`, `n` a nonnegative integer	`1/(n+1)x^(n+1)`
`x^n`, `n` a negative integer `!=-1`     and `x != 0`	`1/(n+1)x^(n+1)`
`x^r`, `r` is a real number and `x > 0`	`1/(r+1)x^(r+1)`
`1/x`, `x > 0`	`ln x`
`1/x`, `x < 0`	`lntext[(]-xtext[)]`
`1/x`, `x != 0`	`ln |x|`
`1/(1+x^2)`	`tan^(-1) x`
`1/sqrt(1-x^2)`, `-1<x<1`	`sin^(-1) x`
`sintext[(]r thetatext[)]`	`-1/r costext[(]r thetatext[)]`
`costext[(]r thetatext[)]`	`1/r sintext[(]r thetatext[)]`
`e^(r theta)`	`-1/r e^(r theta)`
`1/(xtext[(]c-xtext[)])`	`1/c  ln |x|/|c-x|`
`cftext[(]xtext[)]`	`cFtext[(]xtext[)]`, where `Ftext[(]xtext[)]` is an antiderivative for `ftext[(]xtext[)]`
`ftext[(]xtext[)]+gtext[(]xtext[)]`	`Ftext[(]xtext[)]+Gtext[(]xtext[)]`, where `Ftext[(]xtext[)]` is an antiderivative for `ftext[(]xtext[)]` and `Gtext[(]xtext[)]` is an antiderivative for `gtext[(]xtext[)]`

Other formulas

• `1/(xtext[(]c-xtext[)])=1/c[1/x+1/(c-x)]`

• The solution of the logistic growth initial value problem

  `(dP)/(dt)=kPtext[(]M-Ptext[)]` with `Ptext[(]0text[)]=P_0`

  is

  `P=(P_0 M)/(text[(]M-P_0text[)]e^(-Mkt)+P_0)`.