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 = \(1\over t\) 

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