Constant, power and constant factor rule

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Hint

Do not forget to set the constant of integration $\color{purple}{C}$ at the end of indefinite integrals.

Constant rule

If there is only one constant $k$ in the integral, then the integral is:

$\int k\,\mathrm{d}x=kx+C$

Examples

$\int \color{red}{2}\,\mathrm{d}x=\color{red}{2}x\color{purple}{+C}$
$\int \color{red}{5}\,\mathrm{d}x=\color{red}{5}x\color{purple}{+C}$

Power rule

The integral of powers in the form $x^n$ is:

$\int x^n \,\mathrm{d}x=$ $\frac{1}{n+1}x^{n+1}+C$

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Note

$n$ can be any real number except -1, otherwise one would divide by 0.

$n\in\mathbb{R}\backslash\{-1\}$

Examples

$\int x^\color{red}{3} \, \mathrm{d}x $ $= \frac{1}{\color{red}{3}+1}x^{\color{red}{3}+1} $ $= \frac{1}{4}x^{4}\color{purple}{+C}$

$\int x^\color{red}{-2} \, \mathrm{d}x $ $= \frac{1}{\color{red}{-2}+1}x^{\color{red}{-2}+1} $ $= -x^{-1} \color{purple}{+C}$

Constant factor rule

A constant factor can be separated from the integrand and instead multiplied by the integral.

$\int a \cdot g(x) \, \mathrm{d}x =$ $ a \cdot \int g(x) \, \mathrm{d}x$

Examples

Here the power rule and the constant factor rule are applied:

$\int\color{red}{4}x^3 \, \mathrm{d}x$ $=\color{red}{4}\cdot \int x^\color{blue}{3} \,\mathrm{d}x$ $=4\cdot\frac{1}{\color{blue}{3}+1}x^{\color{blue}{3}+1}$ $=x^4\color{purple}{+C}$

$\int\color{red}{-3}x \, \mathrm{d}x$ $=\color{red}{-3}\cdot \int x^\color{blue}{1} \,\mathrm{d}x$ $=-3\cdot\frac{1}{\color{blue}{1}+1}x^{\color{blue}{1}+1}$ $=-\frac32x^2\color{purple}{+C}$

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Note

It is always important to note which variable is integrated.

Example:
$\int 3\,\color{red}{\mathrm{d}y}=3\color{red}{y}+C$