Question

How do you integrate $\dfrac{{{x^3} - 4x - 10}}{{{x^2} - x - 6}}$ using partial fractions?

Answer 1

In this question we have to find the integral of the given function using partial fractions, first perform long division method for partition, then we will get an equation then convert the equation by factoring the denominator into partial fractions, now we will find the values of the variables that we considered, then we will get an expression that is easy to be integrated, so apply integration formula we will get the required value for the given expression.

Complete step by step solution:
Given $\dfrac{{{x^3} - 4x - 10}}{{{x^2} - x - 6}}$ and we have to integrate using partial fractions,
Let,
$I = \int {\dfrac{{{x^3} - 4x - 10}}{{{x^2} - x - 6}}}$,
Now using long division method, divide numerator by denominator, we get,
$\left. { \Rightarrow {x^2} - x - 6} \right){x^3} - 4x - 10\left( x \right. + 1$
$\underline {{x^3} - {x^2} - 6x}$
${x^2} + 2x - 10$
$\,\underline {{x^2} - x - 6}$
$3x - 4$,
Now the given function can be rewritten as,
$\Rightarrow \dfrac{{{x^3} - 4x - 10}}{{{x^2} - x - 6}} = x + 1 + \dfrac{{3x - 4}}{{{x^2} - x - 6}}$,
Now factorise the denominator , we get,
$\Rightarrow \dfrac{{{x^3} - 4x - 10}}{{{x^2} - x - 6}} = x + 1 + \dfrac{{3x - 4}}{{{x^2} - 3x + 2x - 6}}$,
Now simplifying we get,
$\Rightarrow \dfrac{{{x^3} - 4x - 10}}{{{x^2} - x - 6}} = x + 1 + \dfrac{{3x - 4}}{{x\left( {x - 3} \right) + 2\left( {x - 3} \right)}}$,
Again simplifying we get,
Now in the part $\dfrac{{3x - 4}}{{\left( {x + 2} \right)\left( {x - 3} \right)}}$ since the factors are linear then the coefficients of the partial fractions will be constants, say A and B. Writing the function in terms of its partial fractions, we get,
$\Rightarrow \dfrac{{3x - 4}}{{\left( {x - 3} \right)\left( {x + 2} \right)}} = \dfrac{A}{{x - 3}} + \dfrac{B}{{x + 2}}$,
Now taking L.C.M on the right hand side and simplifying we get,
$\Rightarrow \dfrac{{3x - 4}}{{\left( {x - 3} \right)\left( {x + 2} \right)}} = \dfrac{{A\left( {x + 2} \right)}}{{x - 3}} + \dfrac{{B\left( {x - 3} \right)}}{{x + 2}}$,
Now simplifying we get,
$\Rightarrow \dfrac{{3x - 4}}{{\left( {x - 3} \right)\left( {x + 2} \right)}} = \dfrac{{A\left( {x + 2} \right) + B\left( {x - 3} \right)}}{{\left( {x + 2} \right)\left( {x - 3} \right)}}$,
Now further simplifying we get,
$\Rightarrow \dfrac{{3x - 4}}{{\left( {x - 3} \right)\left( {x + 2} \right)}} = \dfrac{{Ax + 2A + Bx - 3B}}{{\left( {x + 2} \right)\left( {x - 3} \right)}}$,
Now taking common terms we get,
$\Rightarrow \dfrac{{3x - 4}}{{\left( {x - 3} \right)\left( {x + 2} \right)}} = \dfrac{{x\left( {A + B} \right) + 2A - 3B}}{{\left( {x + 2} \right)\left( {x - 3} \right)}}$,
Now comparing the two sides we get,
$A + B = 3 - - - - (1)$ and
$2A - 3B = - 4 - - - - (2)$,
Now solving the two equations we get,
$\Rightarrow A = 1$and$B = 2$,
Now substitute the values in the expression we get,
$\Rightarrow \dfrac{{3x - 4}}{{\left( {x - 3} \right)\left( {x + 2} \right)}} = \dfrac{1}{{x - 3}} + \dfrac{2}{{x + 2}}$,
Now the whole expression becomes,
$\Rightarrow \dfrac{{{x^3} - 4x - 10}}{{{x^2} - x - 6}} = x + 1 + \dfrac{1}{{x - 3}} + \dfrac{2}{{x + 2}}$
Now applying integration on both sides we get,
$\Rightarrow I = \int {\dfrac{{{x^3} - 4x - 10}}{{{x^2} - x - 6}}} = \int {x + 1 + \dfrac{1}{{x - 3}} + \dfrac{2}{{x + 2}}}$,
Now separating the terms we get,
$\Rightarrow I = \int {\dfrac{{{x^3} - 4x - 10}}{{{x^2} - x - 6}}} = \int {x + \int 1 + \int {\dfrac{1}{{x - 3}}} + \int {\dfrac{2}{{x + 2}}} }$,
Now applying integration using the integration identities we get,
$\Rightarrow I = \int {\dfrac{{{x^3} - 4x - 10}}{{{x^2} - x - 6}}} = \dfrac{{{x^2}}}{2} + x + \ln \left| {x - 3} \right| + 2\ln \left| {x + 2} \right| + C$,
So, the value of the given expression is,
$\dfrac{{{x^2}}}{2} + x + \ln \left| {x - 3} \right| + 2\ln \left| {x + 2} \right| + C$,

$\therefore$ The value of the given expression $\dfrac{{{x^3} - 4x - 10}}{{{x^2} - x - 6}}$ is equal to $\dfrac{{{x^2}}}{2} + x + \ln \left| {x - 3} \right| + 2\ln \left| {x + 2} \right| + C$.

Note:
Partial fractions is a fraction that can possibly split many fractions. The method of partial fractions basically allows us to split the right hand side of the equation into the left hand side. Partial fractions can be used to turn the functions that cannot be integrated into simple fractions easily.