Math 162

Class 15

Example

\[\int \frac{2 x^{3} + 8 x^2 + 5x}{x^{4} + 3 x^{3} + 2 x^{2} - 2 x - 4}\;dx = \int\frac{1}{x-1}\;dx - \int\frac{1}{x+2}\;dx + \int\frac{2x + 3}{x^2 + 2x + 2}\;dx\]

Continued

\[\int\frac{1}{x^2 + 2x + 2}\;dx\]

Example

\[\int \frac{5x + 3}{3x^2 - 2x + 1}\;dx\]

Example

\[\int \frac{1}{\sqrt{-x^2 + 6x - 5}}\;dx\]

Table gives us:

\[\int \frac{du}{\sqrt{a^2 - u^2}} = \sin^{-1}\frac{u}{a} + C\]

\(\displaystyle \int x^2\sqrt{x^2 + 4x + 13}\;dx\)