Primitive of Reciprocal of x squared by x squared minus a squared squared

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int \frac {\d x} {x^2 \paren {x^2 - a^2}^2} = \frac {-1} {a^4 x} - \frac x {2 a^4 \paren {x^2 - a^2} } + \frac 3 {4 a^5} \map \ln {\frac {x + a} {x - a} } + C$

for $x^2 > a^2$.


Proof

\(\ds \int \frac {\d x} {x^2 \paren {x^2 - a^2}^2}\) \(=\) \(\ds \int \paren {\frac 1 {a^4 x^2} + \frac 3 {4 a^5 \paren {x + a} } - \frac 3 {4 a^5 \paren {x - a} } + \frac 1 {4 a^4 \paren {x + a}^2} + \frac 1 {4 a^4 \paren {x - a}^2} } \rd x\) Partial Fraction Expansion
\(\ds \) \(=\) \(\ds \frac 1 {a^4} \int \frac {\d x} {x^2} + \frac 3 {4 a^5} \int \frac {\d x} {x + a} - \frac 3 {4 a^5} \int \frac {\d x} {x - a} + \frac 1 {4 a^4} \int \frac {\d x} {\paren {x + a}^2} + \frac 1 {4 a^4} \int \frac {\d x} {\paren {x - a}^2}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 1 {a^4} \frac {-1} x + \frac 3 {4 a^5} \int \frac {\d x} {x + a} - \frac 3 {4 a^5} \int \frac {\d x} {x - a} + \frac 1 {4 a^4} \int \frac {\d x} {\paren {x + a}^2} + \frac 1 {4 a^4} \int \frac {\d x} {\paren {x - a}^2} + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {-1} {a^4 x} + \frac 3 {4 a^5} \ln \size {x + a} - \frac 3 {4 a^5} \ln \size {x - a} + \frac 1 {4 a^4} \int \frac {\d x} {\paren {x + a}^2} + \frac 1 {4 a^4} \int \frac {\d x} {\paren {x - a}^2} + C\) Primitive of Reciprocal
\(\ds \) \(=\) \(\ds \frac {-1} {a^4 x} + \frac 3 {4 a^5} \ln \size {x + a} - \frac 3 {4 a^5} \ln \size {x - a} + \frac 1 {4 a^4} \frac {-1} {x + a} + \frac 1 {4 a^4} \frac {-1} {x - a} + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {-1} {a^4 x} + \frac 3 {4 a^5} \ln \size {\frac {x + a} {x - a} } + \frac {-1} {4 a^4 \paren { x + a} } + \frac {-1} {4 a^4 \paren {x - a} } + C\) Difference of Logarithms
\(\ds \) \(=\) \(\ds \frac {-1} {a^4 x} + \frac 3 {4 a^5} \map \ln {\frac {x + a} {x - a} } + \frac {-1} {4 a^4 \paren { x + a} } + \frac {-1} {4 a^4 \paren {x - a} } + C\) Sign of Quotient of Factors of Difference of Squares
\(\ds \) \(=\) \(\ds \frac {-1} {a^4 x} - \frac x {2 a^4 \paren {x^2 - a^2} } + \frac 3 {4 a^5} \map \ln {\frac {x + a} {x - a} } + C\) simplification

$\blacksquare$


Also see


Sources