Primitives of Rational Functions involving a x + b cubed
Jump to navigation
Jump to search
Theorem
This page gathers together the primitives of some rational functions involving $\paren {a x + b}^3$.
Primitive of Reciprocal of $\paren {a x + b}^3$
- $\ds \int \frac {\d x} {\paren {a x + b}^3} = -\frac 1 {2 a \paren {a x + b}^2} + C$
Primitive of $x$ over $\paren {a x + b}^3$
- $\ds \int \frac {x \rd x} {\paren {a x + b}^3} = \frac {-1} {a^2 \paren {a x + b} } + \frac b {2 a^2 \paren {a x + b}^2} + C$
Primitive of $x^2$ over $\paren {a x + b}^3$
- $\ds \int \frac {x^2 \rd x} {\paren {a x + b}^3} = \frac {2 b} {a^3 \paren {a x + b} } - \frac {b^2} {2 a^3 \paren {a x + b}^2} + \frac 1 {a^3} \ln \size {a x + b} + C$
Primitive of $x^3$ over $\paren {a x + b}^3$
- $\ds \int \frac {x^3 \rd x} {\paren {a x + b}^3} = \frac x {a^3} - \frac {3 b^2} {a^4 \paren {a x + b} } + \frac {b^3} {2 a^4 \paren {a x + b}^2} - \frac {3 b} {a^4} \ln \size {a x + b} + C$
Primitive of Reciprocal of $x$ by $\paren {a x + b}^3$
- $\ds\int \frac {\d x} {x \paren {a x + b}^3} = \frac {a^2 x^2} {2 b^3 \paren {a x + b}^2} - \frac {2 a x} {b^3 \paren {a x + b} } + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C$
Primitive of Reciprocal of $x^2$ by $\paren {a x + b}^3$
- $\ds \int \frac {\d x} {x^2 \paren {a x + b}^3} = \frac {-a} {2 b^2 \paren {a x + b}^2} - \frac {2 a} {b^3 \paren {a x + b} } - \frac 1 {b^3 x} + \frac {3 a} {b^4} \ln \size {\frac {a x + b} x} + C$
Primitive of Reciprocal of $x^3$ by $\paren {a x + b}^3$
- $\ds \int \frac {\d x} {x^3 \paren {a x + b}^3} = \frac {a^4 x^2} {2 b^5 \paren {a x + b}^2} - \frac {4 a^3 x} {b^5 \paren {a x + b} } - \frac {\paren {a x + b}^2} {2 b^5 x^2} + \frac {4 a} {b^4 x} + \frac {6 a^2} {b^5} \ln \size {\frac x {a x + b} } + C$