Basic Integration – user's Blog!

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 a d x = a x + C \\ Example \\ \int 2 d x = 2 x + C \end{array}

Type 2:

\begin{array}{l} \int a x^{n} d x = \frac{a x^{n + 1}}{n + 1} + C \\ E x a m p l e 1 \\ \int 2 x^{3} d x = \frac{2 x^{4}}{4} + C = \frac{x^{4}}{2} + C \\ E x a m p l e 2 \\ \int \frac{2}{3 x^{5}} d x = \int \frac{2}{3} x^{- 5} d x = \frac{2}{3} (\frac{x^{- 4}}{- 4}) + C \\ = \frac{2}{3} (\frac{x^{- 4}}{- 4}) + C = \frac{x^{- 4}}{- 6} + C \end{array}

Type 3:

\begin{array}{l} \int (u + v) d x = \int u d x \pm \int v d x \\ u and v are functions in x \\ E x a m p l e 1 \\ \int 3 x^{2} + 2 x d x = \int 3 x^{2} d x + \int 2 x d x \\ = \frac{3 x^{3}}{3} + \frac{2 x^{2}}{2} + C \\ = \frac{3 x^{3}}{3} + \frac{2 x^{2}}{2} + C \\ = x^{3} + x^{2} + C \end{array}

\begin{array}{l} E x a m p l e 2 \\ \int (x + 2) (3 x + 1) d x = \int 3 x^{2} + 7 x + 2 d x \\ = \int 3 x^{2} d x + \int 7 x d x + \int 2 d x \\ = \frac{3 x^{3}}{3} + \frac{7 x^{2}}{2} + 2 x + C \\ = x^{3} + \frac{7 x^{2}}{2} + 2 x + C \end{array}

\begin{array}{l} E x a m p l e 3 \\ \int \frac{3 x^{3} + x^{2} - x}{x} d x = \int 3 x^{2} + x - 1 d x \\ = \int 3 x^{2} d x + \int x d x - \int 1 d x \\ = \frac{3 x^{3}}{3} + \frac{x^{2}}{2} - x + C \\ = x^{3} + \frac{x^{2}}{2} - x + C \end{array}