Primitive of Reciprocal of x cubed by a x + b

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int \frac {\d x} {x^3 \paren {a x + b} } = \frac {2 a x - b} {2 b^2 x^2} + \frac {a^2} {b^3} \ln \size {\frac x {a x + b} } + C$


Proof 1

\(\ds \int \frac {\d x} {x^3 \paren {a x + b} }\) \(=\) \(\ds \int \paren {\frac {a^2} {b^3 x} + \frac {-a} {b^2 x^2} + \frac 1 {b x^3} + \frac {-a^3} {b^3 \paren {a x + b} } } \rd x\) Partial Fraction Expansion
\(\ds \) \(=\) \(\ds \frac {a^2} {b^3} \int \frac {\d x} x + \frac {-a} {b^2} \int \frac {\d x} {x^2} + \frac 1 b \int \frac {\d x} {x^3} + \frac {-a^3} {b^3} \int \frac {\d x} {a x + b}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac {a^2} {b^3} \int \frac {\d x} x + \frac {-a} {b^2} \frac {-1} x + \frac 1 b \frac {-1} {2 x^2} + \frac {-a^3} {b^3} \int \frac {\d x} {a x + b} + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {a^2} {b^3} \ln \size x + \frac a {b^2 x} - \frac 1 {2 b x^2} + \frac {-a^3} {b^3} \int \frac {\d x} {a x + b} + C\) Primitive of Reciprocal
\(\ds \) \(=\) \(\ds \frac {a^2} {b^3} \ln \size x + \frac a {b^2 x} - \frac 1 {2 b x^2} + \frac {-a^3} {b^3} \frac 1 a \ln \size {a x + b} + C\) Primitive of $\dfrac 1 {a x + b}$
\(\ds \) \(=\) \(\ds \frac a {b^2 x} - \frac 1 {2 b x^2} + \frac {a^2} {b^3} \ln \size {\frac x {a x + b} } + C\) Difference of Logarithms
\(\ds \) \(=\) \(\ds \frac {2 a x - b} {2 b^2 x^2} + \frac {a^2} {b^3} \ln \size {\frac x {a x + b} } + C\) rearranging

$\blacksquare$


Proof 2

\(\ds \int \frac {\d x} {x^3 \paren {a x + b} }\) \(=\) \(\ds \int \frac {b \rd x} {b x^3 \paren {a x + b} }\) multiplying top and bottom by $b$
\(\ds \) \(=\) \(\ds \int \frac {\paren {a x + b - a x} \rd x} {b x^3 \paren {a x + b} }\) adding and subtracting $a x$
\(\ds \) \(=\) \(\ds \frac 1 b \int \frac {\paren {a x + b} \rd x} {x^3 \paren {a x + b} } - \frac a b \int \frac {x \rd x} {x^3 \paren {a x + b} }\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 1 b \int \frac {\d x} {x^3} - \frac a b \int \frac {\d x} {x^2 \paren {a x + b} }\) simplification
\(\ds \) \(=\) \(\ds -\frac 1 {2 b x^2} - \frac a b \int \frac {\d x} {x^2 \paren {a x + b} } + C\) Primitive of Power
\(\ds \) \(=\) \(\ds -\frac 1 {2 b x^2} - \frac a b \paren {-\frac 1 {b x} + \frac a {b^2} \ln \size {\frac {a x + b} x} } + C\) Primitive of $\dfrac 1 {x^2 \paren {a x + b} }$
\(\ds \) \(=\) \(\ds -\frac 1 {2 b x^2} + \frac a {b^2 x} - \frac {a^2} {b^3} \ln \size {\frac {a x + b} x} + C\) multiplying out
\(\ds \) \(=\) \(\ds \frac a {b^2 x} - \frac 1 {2 b x^2} + \frac {a^2} {b^3} \ln \size {\frac x {a x + b} } + C\) Difference of Logarithms
\(\ds \) \(=\) \(\ds \frac {2 a x - b} {2 b^2 x^2} + \frac {a^2} {b^3} \ln \size {\frac x {a x + b} } + C\) rearranging

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Primitive_of_Reciprocal_of_x_cubed_by_a_x_%2B_b&oldid=603349"