Definition
Indefinite Integral
\(\samefrac{df(x)}{dx}=G(x)\)
\(\Rightarrow\displaystyle\int G(x)\,dx=f(x)+c\)
Example
\((x^2)'=2x\)
\((x^2+1)'=2x\)
\((x^2+3)'=2x\)
\((x^2+c)'=2x\qquad c\in\mathbb R\)
\(\displaystyle\int(2x)\,dx=x^2+c\)
Property 1
\(\blacksquare\quad\displaystyle\int x^n\,dx=\samefrac{x^{n+1}}{n+1}+c\qquad n\ne-1\)
\(\blacksquare\quad\displaystyle\int\samefrac1x\,dx=\ln|x|+c\)
1.
\(\displaystyle\int x^2\,dx=?\)
2.
\(\displaystyle\int x^4\,dx=?\)
3.
\(\displaystyle\int x\,dx=?\)
4.
\(\displaystyle\int \samefrac1{x^2}\,dx=?\)
5.
\(\displaystyle\int \sqrt x\,dx=?\)
6.
\(\displaystyle\int \sqrt[3] x\,dx=?\)
7.
\(\displaystyle\int y^{-4}\,dy=?\)
8.
\(\displaystyle\int u^2\,du=?\)
9.
\(\displaystyle\int \sqrt{x^3}\,dx=?\)
10.
\(\displaystyle\int dx=?\)
11.
\(\displaystyle\int \samefrac{x\cdot\sqrt x}{\sqrt[4] x}\,dx=?\)
12.
\(\displaystyle\int \sqrt[4]{x^7}\,dx=?\)
13.
\(\displaystyle\int \sqrt[6]{x^{-2}}\,dx=?\)
14.
\(\displaystyle\int x^{-5}\,dx=?\)