Saved Bookmarks
| 1. |
\[ \frac{x^{2}-x+1}{\left(x^{2}+1\right)^{3 / 2}} \] integrate this function. |
|
Answer» \(\int \frac{x^2 - x + 1}{(x^2 + 1)^{3/2}}\, dx = \int \frac{x^2 + 1}{(x^2 + 1)^{3/2}}\,dx - \frac 12 \int \frac{2x}{(x^2 + 1)^{3/2}} \,dx\) \(= \int \frac{1}{\sqrt{1+ x^2}}\,dx - \frac12 \int \frac{dt}{t^{3/2}}\) (By taking \(x^2 + 1 = t\)) \(= log|x+ \sqrt{1 + x^2}| - \frac 12 \times \frac{t^{\frac{-3}2 + 1}}{\frac{-3}2 + 1} + C\) \(= log |x + \sqrt{1+ x^2}| + \frac 1{t^{1/2}} + C\) \(= log|x + \sqrt{1 + x^2}| + \frac1{\sqrt{1 + x^2}} + C\). |
|