Primitive of Reciprocal of x by a x + b cubed

From ProofWiki
Jump to navigation Jump to search

Theorem

$\displaystyle \int \frac {\mathrm d x} {x \left({a x + b}\right)^3} = \frac {a^2 x^2} {2 b^3 \left({a x + b}\right)^2} - \frac {2 a x} {b^3 \left({a x + b}\right)} + \frac 1 {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C$


Proof

\(\displaystyle \int \frac {\mathrm d x} {x \left({a x + b}\right)^3}\) \(=\) \(\displaystyle \int \left({\frac 1 {b^3 x} - \frac a {b^3 \left({a x + b}\right)} - \frac a {b^2 \left({a x + b}\right)^2} - \frac a {b \left({a x + b}\right)^3} }\right) \ \mathrm d x\) Partial Fraction Expansion
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b^3} \int \frac {\mathrm d x} x - \frac a {b^3} \int \frac {\mathrm d x} {a x + b} - \frac a {b^2} \int \frac {\mathrm d x} {\left({a x + b}\right)^2} - \frac a b \int \frac {\mathrm d x} {\left({a x + b}\right)^3}\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b^3} \ln \left\vert{x}\right\vert - \frac a {b^3} \int \frac {\mathrm d x} {a x + b} - \frac a {b^2} \int \frac {\mathrm d x} {\left({a x + b}\right)^2} - \frac a b \int \frac {\mathrm d x} {\left({a x + b}\right)^3} + C\) Primitive of Reciprocal
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b^3} \ln \left\vert{x}\right\vert - \frac 1 {b^3} \ln \left\vert{a x + b}\right\vert - \frac a {b^2} \int \frac {\mathrm d x} {\left({a x + b}\right)^2} - \frac a b \int \frac {\mathrm d x} {\left({a x + b}\right)^3} + C\) Primitive of Reciprocal of a x + b
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b^3} \ln \left\vert{x}\right\vert - \frac 1 {b^3} \ln \left\vert{a x + b}\right\vert - \frac a {b^2} \frac {-1} {a \left({a x + b}\right)} - \frac a b \int \frac {\mathrm d x} {\left({a x + b}\right)^3} + C\) Primitive of Reciprocal of a x + b squared
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b^3} \ln \left\vert{x}\right\vert - \frac 1 {b^3} \ln \left\vert{a x + b}\right\vert - \frac a {b^2} \frac {-1} {a \left({a x + b}\right)} - \frac a b \frac {-1} {2 a \left({a x + b}\right)^2} + C\) Primitive of Reciprocal of a x + b cubed
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b^2 \left({a x + b}\right)} + \frac 1 {2 b \left({a x + b}\right)^2} + \frac 1 {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C\) Difference of Logarithms and rearranging
\(\displaystyle \) \(=\) \(\displaystyle \frac {b} {b^3 \left({a x + b}\right)} + \frac {b^2} {2 b^3 \left({a x + b}\right)^2} + \frac 1 {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C\)
\(\displaystyle \) \(=\) \(\displaystyle \left({ \frac {a x + b} {b^3 \left({a x + b}\right)} - \frac {a x} {b^3 \left({a x + b}\right)} }\right) + \left({ \frac {a^2 x^2 + 2 a b x + b^2} {2 b^3 \left({a x + b}\right)^2} - \frac {a^2 x^2 + 2 a b x} {2 b^3 \left({a x + b}\right)^2} }\right) + \frac 1 {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {a x + b} {b^3 \left({a x + b}\right)} - \frac {a x} {b^3 \left({a x + b}\right)} + \frac {\left({a x + b}\right)^2} {2 b^3 \left({a x + b}\right)^2} - \frac {a^2 x^2 + 2 a b x} {2 b^3 \left({a x + b}\right)^2} + \frac 1 {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b^3} - \frac {a x} {b^3 \left({a x + b}\right)} + \frac 1 {2 b^3} - \frac {a^2 x^2 + 2 a b x} {2 b^3 \left({a x + b}\right)^2} + \frac 1 {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C\)
\(\displaystyle \) \(=\) \(\displaystyle - \frac {a x} {b^3 \left({a x + b}\right)} - \frac {a^2 x^2 + 2 a b x} {2 b^3 \left({a x + b}\right)^2} + \frac 1 {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C + \frac 3 {2 b^3}\)
\(\displaystyle \) \(=\) \(\displaystyle - \frac {a x} {b^3 \left({a x + b}\right)} - \frac {a^2 x^2} {2 b^3 \left({a x + b}\right)^2} - \frac {2 a b x} {2 b^3 \left({a x + b}\right)^2} + \frac 1 {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C + \frac 3 {2 b^3}\)
\(\displaystyle \) \(=\) \(\displaystyle - \frac {a x} {b^3 \left({a x + b}\right)} - \frac {a^2 x^2} {2 b^3 \left({a x + b}\right)^2} - \frac {2 a x \left({a x + b}\right)} {2 b^3 \left({a x + b}\right)^2} + \frac {2 a^2 x^2} {2 b^3 \left({a x + b}\right)^2} + \frac 1 {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C + \frac 3 {2 b^3}\)
\(\displaystyle \) \(=\) \(\displaystyle - \frac {a x} {b^3 \left({a x + b}\right)} - \frac {a^2 x^2} {2 b^3 \left({a x + b}\right)^2} - \frac {2 a x} {2 b^3 \left({a x + b}\right)} + \frac {2 a^2 x^2} {2 b^3 \left({a x + b}\right)^2} + \frac 1 {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C + \frac 3 {2 b^3}\)
\(\displaystyle \) \(=\) \(\displaystyle - \frac {a x} {b^3 \left({a x + b}\right)} + \frac {a^2 x^2} {2 b^3 \left({a x + b}\right)^2} - \frac {a x} {b^3 \left({a x + b}\right)} + \frac 1 {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C + \frac 3 {2 b^3}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {a^2 x^2} {2 b^3 \left({a x + b}\right)^2} - \frac {2 a x} {b^3 \left({a x + b}\right)} + \frac 1 {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C + \frac 3 {2 b^3}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {a^2 x^2} {2 b^3 \left({a x + b}\right)^2} - \frac {2 a x} {b^3 \left({a x + b}\right)} + \frac 1 {b^3} \ln \left\vert{\frac x {a x + b} }\right\vert + C\) $C + \frac 3 {2 b^3}$ is constant

$\blacksquare$


Sources