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Line 13: + − + − + − :<big><math>\int\frac{3x+11}{x^2-x-6}dx</math></big>+ − + − The first step is to factor the denominator as much as possible and get the form of the partial fraction decomposition. Doing this gives,+ − − :<big><math>\frac{3x+11}{(x-3)(x+2)}=\frac{A}{x-3}+\frac{B}{x+2}</math></big> − − This allows us to split the fraction in to sums by cross multiplying the denominators, − − :<big><math>\frac{3x+11}{(x-3)(x+2)}=\frac{A(x+2)+B(x-3)}{(x-3)(x+2)}</math></big> − − Therefore we can restate the problem, − − :<big><math>3x+11=A(x+2)+B(x-3)</math></big> − − Now we can solve for A and B by subsituting x with a value that allows the term to go to 0. For example, − − We let :<big><math>x=-2</math></big>, − − :<big><math>3(-2)+11=A(-2+2)+B(-2-3)</math></big> − :<big><math>5=A(0)+B(5)</math></big> − :<big><math>B=-1</math></big> − − We let :<big><math>x=3</math></big>, − − :<big><math>3(3)+11=A(3+2)+B(3-3)</math></big> − :<big><math>20=A(5)+B(0)</math></big> − :<big><math>A=4</math></big> − − We then plug in the values of A and B and get, − − :<big><math>\frac{4}{x-3}-\frac{1}{x+2}</math></big> − − Now we can solve the integral. − − :<big><math>\int\frac{3x+11}{x^2-x-6}dx=\int\frac{4}{x-3}-\frac{1}{x+2}dx</math></big> − :<big><math>\int\frac{3x+11}{x^2-x-6}dx=\int\frac{4}{x-3}dx-\int\frac{1}{x+2}dx</math></big> − :<big><math>\int\frac{3x+11}{x^2-x-6}dx=4ln|x-3|-ln|x+2|+c</math></big>
Methods of integration (view source)
Revision as of 13:50, December 30, 2008
, 13:50, December 30, 2008→Partial Fractions: moved majority of material to Partial fractions in integration
==Partial Fractions==
==Partial Fractions==
{{Main|Partial fractions in integration}}
'''Integration by partial fractions''' is a [[Techniques of integration|technique]] to facilitate the integration of a rational expression by partial fraction decomposition.
'''Integration by partial fractions''' is a [[Techniques of integration|technique]] to facilitate the integration of a rational expression by partial fraction decomposition.
Given an integral
Given an integral
<br />
<math>\int \frac {f(x)}{g(x)}dx</math>
<br />
where <math>f(x)</math> and <math>g(x)</math> are both polynomials, integration by partial fractions shows how to separate the problem into multiple integrals before integrating.
==Algebraic Substitution==
==Algebraic Substitution==