Primitive of Reciprocal of a squared minus x squared squared
Primitive of Reciprocal of a squared minus x squared squared 0 34060 188713 2014-06-06T22:10:40Z Prime.mover 59 Created page with "== Theorem ==  :$\displaystyle \int \frac {\mathrm d x} {\left({a^2 - x^2}\right)^2} = \frac x {2 a^2 \left({a^2 - x^2}\right)} + \frac 1 {4 a^3} \ln \left({\frac..." wikitext text/x-wiki == Theorem ==  :$\displaystyle \int \frac {\mathrm d x} {\left({a^2 - x^2}\right)^2} = \frac x {2 a^2 \left({a^2 - x^2}\right)} + \frac 1 {4 a^3} \ln \left({\frac {a + x} {a - x} }\right) + C$ for $x^2 < a^2$.  == Proof == {{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {\left({a^2 - x^2}\right)^2}       | r = \int \left({\frac 1 {4 a^3 \left({a + x}\right)} + \frac 1 {4 a^3 \left({a - x}\right)} + \frac 1 {4 a^2 \left({a + x}\right)^2} + \frac 1 {4 a^2 \left({a - x}\right)^2} }\right) \ \mathrm d x       | c = Partial Fraction Expansion }} {{eqn | r = \frac 1 {4 a^3} \int \frac {\mathrm d x} {a + x} + \frac 1 {4 a^3} \int \frac {\mathrm d x} {a - x} + \frac 1 {4 a^2} \int \frac {\mathrm d x} {\left({a + x}\right)^2} + \frac 1 {4 a^2} \int \frac {\mathrm d x} {\left({a - x}\right)^2}       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 {4 a^3} \ln \left\vert{a + x}\right\vert - \frac 1 {4 a^3} \ln \left\vert{a - x}\right\vert + \frac 1 {4 a^2} \int \frac {\mathrm d x} {\left({a + x}\right)^2} + \frac 1 {4 a^2} \int \frac {\mathrm d x} {\left({a - x}\right)^2} + C       | c = Primitive of Function of $a x + b$ and Primitive of Reciprocal }} {{eqn | r = \frac 1 {4 a^3} \ln \left\vert{\frac {a + x} {a - x} }\right\vert + \frac 1 {4 a^2} \int \frac {\mathrm d x} {\left({a + x}\right)^2} + \frac 1 {4 a^2} \int \frac {\mathrm d x} {\left({a - x}\right)^2} + C       | c = Difference of Logarithms }} {{eqn | r = \frac 1 {4 a^3} \ln \left({\frac {a + x} {a - x} }\right) + \frac 1 {4 a^2} \int \frac {\mathrm d x} {\left({a + x}\right)^2} + \frac 1 {4 a^2} \int \frac {\mathrm d x} {\left({a - x}\right)^2} + C       | c = Sign of Quotient of Factors of Difference of Squares }} {{eqn | r = \frac 1 {4 a^3} \ln \left({\frac {a + x} {a - x} }\right) + \frac 1 {4 a^2} \frac {-1} {\left({a + x}\right)} + \frac 1 {4 a^2} \frac 1 {\left({a - x}\right)} + C       | c = Primitive of Function of $a x + b$ and Primitive of Power }} {{eqn | r = \frac 1 {4 a^3} \ln \left({\frac {a + x} {a - x} }\right) + \left({\frac 1 {4 a^2 \left({a - x}\right)} - \frac 1  {4 a^2 \left({a + x}\right)} }\right) + C       | c = simplifying }} {{eqn | r = \frac 1 {4 a^3} \ln \left({\frac {a + x} {a - x} }\right) + \left({\frac {\left({a + x}\right) - \left({a - x}\right)} {4 a^2 \left({a - x}\right) \left({a + x}\right)} }\right) + C       | c = simplifying }} {{eqn | r = \frac x {2 a^2 \left({a^2 - x^2}\right)} + \frac 1 {4 a^3} \ln \left({\frac {a + x} {a - x} }\right) + C       | c = Difference of Two Squares }} {{end-eqn}}  == Also see == * Primitive of $\dfrac 1 {\left({x^2 - a^2}\right)^2}$ == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Primitive of Reciprocal of x cubed by a squared minus x squared|next=Primitive of x over a squared minus x squared squared}}: $\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.170$