# How can I resolve rational indefinite integral?

$$\int\left( {20.56\over x^2-1.27}+x^{55}\right) dx$$

I came to something like this

$$20.56 \int {1\over (x-1)^2 - \frac{27}{100} }~dx + {x^{56}\over56} + \text{Constant}$$

Please can you help me to resolve this?

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Your integral is of the form $$I = \int \left(\dfrac{a}{x^2 - b^2} + x^n \right)dx$$ Hence, \begin{align} I & = \int \left(\dfrac{a}{2b}\dfrac1{x - b} - \dfrac{a}{2b}\dfrac1{x + b} + x^n \right)dx \\ & = \dfrac{a}{2b} \log(\vert x-b \vert) - \dfrac{a}{2b} \log(\vert x+b \vert) + \dfrac{x^{n+1}}{n+1} + \text{constant}\\ & = \dfrac{a}{2b} \log\left(\left \vert \dfrac{x-b}{x+b} \right \vert \right) + \dfrac{x^{n+1}}{n+1} + \text{constant} \end{align} In your case, $a = 20.56$, $b = \sqrt{1.27}$ and $n = 55$.

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thanks a lot sir – Sam Dec 16 '12 at 19:21