Table - rational functions
Task number: 3110
Fill in the following table (determine the intervals on which the solution is valid). Express the last function using integration by parts with a recurrence relation in terms of \(\int \frac{1}{(x^2 + 1)^k} \, dx\).
| \(f(x)\) | \(F(x)\) |
|---|---|
| \(\frac{1}{x - \alpha}\) | |
| \(\frac{1}{(x - \alpha)^k}\); \(k > 1\) | |
| \(\frac{2x + p}{x^2 + px + q}\) | |
| \(\frac{1}{x^2 + px + q}\); \(q > \frac{p^2}4\) | |
| \(\frac{2x + p}{(x^2 + px + q)^k}\); \(k>1\) | |
| \(\frac{1}{(x^2 + 1)^{k+1}}\); \(k \geq 1\) |
Result
To simplify the notation, the solutions are written without the constant.
\(f(x)\) \(F(x)\) \(\frac{1}{x - \alpha}\) \(\ln|x - \alpha|\) \(\frac{1}{(x - \alpha)^k}\); \(k > 1\) \(\frac{-1}{(k-1)(x - \alpha)^{k-1}}\) \(\frac{2x + p}{x^2 + px + q}\) \(\ln|x^2 + px + q|\) \(\frac{1}{x^2 + px + q}\); \(q > \frac{p^2}4\) \(\frac{1}{\sqrt t} \operatorname{arctan} ((x + p/2)/\sqrt t)\), where \(t = q - p^2/4\) \(\frac{2x + p}{(x^2 + px + q)^k}\); \(k>1\) \(\frac{-1}{(k-1)(x^2 + px + q)^{k-1}}\) \(\frac{1}{(x^2 + 1)^{k+1}}\); \(k \geq 1\) \(\frac{1}{2k} \left(\frac{x}{(1+x^2)^k} + (2k-1) \int \frac{1}{(x^2 + 1)^k} \, dx \right)\)
