3.1 Basic Integration

3.1 Integration as the Inverse of Differentiation, Integration of axn and integration of the Functions of the Sum/Difference of Algebraic Terms

Type 1:

$\begin{array}{l}\int adx=ax+C\\ \text{Example}\\ \int 2dx=2x+C\end{array}$

Type 2:

$\begin{array}{l}\int a{x}^{n}dx=\frac{a{x}^{n+1}}{n+1}+C\\ \\ \\ Example\text{1}\\ \int 2{x}^{3}dx=\frac{2{x}^{4}}{4}+C=\frac{{x}^{4}}{2}+C\\ \\ \\ Example\text{2}\\ \int \frac{2}{3{x}^{5}}dx=\int \frac{2}{3}{x}^{-5}dx=\frac{2}{3}\left(\frac{{x}^{-4}}{-4}\right)+C\\ =\frac{\overline{)2}}{3}\left(\frac{{x}^{-4}}{\overline{)-4}}\right)+C=\frac{{x}^{-4}}{-6}+C\end{array}$

Type 3:

$\begin{array}{l}Example\text{2}\\ \int \left(x+2\right)\left(3x+1\right)dx=\int 3{x}^{2}+7x+2dx\\ =\int 3{x}^{2}dx+\int 7xdx+\int 2dx\\ =\frac{3{x}^{3}}{3}+\frac{7{x}^{2}}{2}+2x+C\\ ={x}^{3}+\frac{7{x}^{2}}{2}+2x+C\end{array}$

$\begin{array}{l}Example\text{3}\\ \int \frac{3{x}^{3}+{x}^{2}-x}{x}dx=\int 3{x}^{2}+x-1dx\\ =\int 3{x}^{2}dx+\int xdx-\int 1dx\\ =\frac{3{x}^{3}}{3}+\frac{{x}^{2}}{2}-x+C\\ ={x}^{3}+\frac{{x}^{2}}{2}-x+C\end{array}$