Integrate the rational function: \(\frac {2}{(1-x)(1+x^2)}\)
Solution and Explanation
Let \(\frac {2}{(1-x)(1+x^2)}\) \(=\) \(\frac {A}{(1-x)}+\frac {Bx+C}{(1+x^2)}\)
\(2 = A(1+x^2)+(Bx+C)(1-x)\)
\(2 = A+Ax^2+Bx-Bx+C-Cx\)
Equating the coefficient of x2, x, and constant term, we obtain
\(A − B = 0\)
\(B − C = 0\)
\(A + C = 2\)
On solving these equations, we obtain
\(A = 1, \ B = 1, \ and \ C = 1\)
∴ \(\frac {2}{(1-x)(1+x^2)}\) = \(\frac {1}{1-x}\) + \(\frac {x+1}{1+x^2}\)
⇒ \(∫\)\(\frac {2}{(1-x)(1+x^2)}\) = \(∫\)\(\frac {1}{1-x}\ dx\)+ \(∫\)\(\frac {x}{1+x^2}\ dx\) + \(∫\)\(\frac {1}{1+x^2}\ dx\)
= - \(∫\)\(\frac {1}{1-x}\ dx\) + \(\frac 12\)\(∫\)\(\frac {2x}{1+x^2}\ dx\) + \(∫\)\(\frac {1}{1+x^2}\ dx\)
= -\(log\ |x-1|+\frac 12log|1+x^2|+tan^{-1}x+C\)
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Concepts Used:
Integration by Partial Fractions
The number of formulas used to decompose the given improper rational functions is given below. By using the given expressions, we can quickly write the integrand as a sum of proper rational functions.
For examples,
