Integration by Partial Fraction Formula

Here you will learn integration by partial fraction formula and integration of irrational functions.

Let’s begin –

Integration by Partial Fraction Formula

(i) Integration of Rational Functions

S.No	form of rational function	form of partial fraction
1	$px^2+qx+r\over {(x-a)(x-b)(x-c)}$	$A\over {x-a}$ + $B\over {x-b}$ + $C\over {x-c}$
2	$px^2+qx+r\over {{(x-a)}^2(x-b)}$	$A\over {x-a}$ + $B\over {(x-a)}^2$ + $C\over {x-b}$
3	$px^2+qx+r\over {(x-a)(x^2+bx+c)}$	$A\over {x-a}$ + $Bx+C\over {x^2+bx+c}$

Example : Evaluate $\int$ $x\over {(x-2)(x-5)}$ dx

Solution : We have, $\int$ $x\over {(x-2)(x-5)}$ dx

Let $x\over {(x-2)(x-5)}$ = $A\over {x-2}$ + $B\over {x-5}$

or   x = A(x+5) + B(x-2)

by comparing the coefficients, we get

A = 2/7 and B = 5/7 so that

$\int$ $x\over {(x-2)(x-5)}$ dx = $2\over 7$ $\int$$dx\over x-2$ + $5\over 7$ $\int$$dx\over x+5$

= $2\over 7$ ln|x-2| + $5\over 7$ ln|x+5| + C

Example : Evaluate $\int$ $2x\over {(x^2+1)(x^2+2)}$ dx

Solution : Let I = $\int$ $2x\over {(x^2+1)(x^2+2)}$ dx

Putting $x^2$ = t and 2xdx = dt, we get

I = $\int$ $dt\over {(t+1)(t+2)}$

Let $1\over {(t+1)(t+2)}$ = $A\over t+1$ + $B\over t+2$ …….(i)

$\implies$ 1 = A(t+2) + B(t+1) ……..(ii)

Putting t = -2 in (ii), we obtain B = -1

Putting t = -1 in (ii), we obtain A = 1

Putting value of A and B in (i), we get

$1\over {(t+1)(t+2)}$ = $1\over t+1$ – $1\over t+2$

I = $\int$ $1\over {(t+1)(t+2)}$

$\implies$ I = $\int$ $1\over t+1$dt – $\int$ $1\over t+2$dt

$\implies$ I = log|t+1| – log|t+2| + C

$log|x^2+1|$ – $log|x^2+2|$ + C

(ii)  Integration of Irrational Functions

(a) $\int$ $dx\over {(ax + b)\sqrt{px+q}}$ & $\int$ $dx\over {(ax^2 + bx + c)\sqrt{px+q}}$; put px+q = $t^2$

(b)  $\int$ $dx\over {(ax + b)\sqrt{px^2+qx+r}}$; put ax+b = $1\over t$; $\int$ $dx\over {(ax^2 + b)\sqrt{px^2+q}}$; put x = $1\over t$