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 = \(1\over t\)