Integral

$\int x^{n} d x = \frac{1}{n + 1} . x^{n + 1} + c$

$\int (a x + b)^{n} d x = \frac{1}{a} . \frac{1}{n + 1} . x^{n + 1} + c$

$\int k d x = k x + c$ , $k$ is a constant

Example 01

$\begin{aligned} \int x^{4} d x & = \frac{1}{4 + 1} . x^{4 + 1} + c \\ \int x^{4} d x & = \frac{1}{5} x^{5} + c \end{aligned}$

Example 02

$\begin{aligned} \int 3 x^{8} d x & = 3 . \frac{1}{8 + 1} . x^{8 + 1} + c \\ \int 3 x^{8} d x & = \frac{1}{3} x^{9} + c \end{aligned}$

Example 03

$\begin{aligned} \int \sqrt{x} d x & = \int x^{\frac{1}{2}} d x \\ \int \sqrt{x} d x & = \frac{1}{\frac{1}{2} + 1} . x^{\frac{1}{2} + 1} + c \\ \int \sqrt{x} d x & = \frac{1}{\frac{3}{2}} . x^{\frac{1}{2} + 1} + c \\ \int \sqrt{x} d x & = \frac{2}{3} x^{1 \frac{1}{2}} + c \\ \int \sqrt{x} d x & = \frac{2}{3} x \sqrt{x} + c \end{aligned}$

Example 04

$\begin{aligned} \int \frac{1}{x^{3}} d x & = \int x^{- 3} d x \\ \int \frac{1}{x^{3}} d x & = \frac{1}{- 3 + 1} . x^{- 3 + 1} + c \\ \int \frac{1}{x^{3}} d x & = - \frac{1}{2} x^{- 2} + c \\ \int \frac{1}{x^{3}} d x & = - \frac{1}{2 x^{2}} + c \end{aligned}$

Example 05

$\begin{aligned} \int (2 x - 10)^{3} d x \\ \frac{1}{2} . \frac{1}{3 + 1} . (2 x - 10)^{3 + 1} + c \\ \frac{1}{8} (2 x - 10)^{4} + c \end{aligned}$

Example 06

$\begin{aligned} \int \frac{d x}{(4 x - 1)^{3}} \\ \int (4 x - 1)^{- 3} d x \\ \frac{1}{4} . \frac{1}{- 3 + 1} . (4 x - 1)^{- 3 + 1} + c \\ - \frac{1}{8} (4 x - 1)^{- 2} + c \\ \frac{- 1}{8 . (4 x - 1)^{2}} + c \end{aligned}$

Example 07

$\begin{aligned} \int 4 d x & = 4 x + c \end{aligned}$

Exercise

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Integral

Exercise

Integration