Class 13
\(\displaystyle \int_0^1 \frac{1}{x^2 - 4}\;dx\)
\(\displaystyle \int \frac{1}{x^k}\;dx = \begin{cases} \frac{x^{-k+1}}{-k+1} + C_1 & \text{if } x < 0\\ \frac{x^{-k+1}}{-k+1} + C_2 & \text{if } x > 0\end{cases}\)
\(\displaystyle \int \frac{1}{x}\;dx = \begin{cases} \ln(-x) + C_1 & \text{if } x < 0\\ \ln x + C_2 & \text{if } x > 0\end{cases}\)
\(\displaystyle \int \frac{1}{1 + z^2}\;dz = \arctan z + C_1 = -\operatorname{arccot} z + C_2\)
\(\displaystyle \int \frac{x}{1 + x^2}\;dx\)
\(\displaystyle \int \frac{x}{(1 + x^2)^k}\;dx\)
Degree of the numerator is less than the degree of the denominator.
What is a mixed function?
\(\displaystyle \frac{2x^5 + 7x^4 + 3x^3 - 2x^2 + 10x}{x^2 + 2x - 1}\)
\(\displaystyle \frac{5 x^{6} - 4 x^{5} + 7 x^{4} + 8 x^{3} - 5 x^{2} + 6 x + 5}{x^2 - x + 2}\)
\(\displaystyle \frac{x}{x^2 + 5x + 6}\)
\(\displaystyle \frac{3}{(x-1)(x+1)(x-2)}\)
\(\displaystyle \frac{2x+1}{(x+1)(x+2)^2}\)
\(\displaystyle \frac{x^2 + 2x+1}{(x+1)(x^2+2)}\)
\(\displaystyle \frac{x^2 + 2x+1}{(x+1)^3(x^2+2)^2}\)