Math Integration rules Integration of fractions and roots

# Integration of fractions and roots

You can often integrate fractions and roots by first applying the exponent rules and then the integration rules.

!

### Remember

Fractions can be rewritten as a power with a negative exponent:
$\frac{1}{a^x}=a^{-x}$

Roots can also be written as a power with rational exponent:
$\sqrt[n]{a^m}=a^{\frac{m}{n}}$
i

### Method

1. Transform fraction or root into power
2. Apply integration rules
3. If necessary, write power again as a fraction or root

### Examples

$\int \frac{1}{x^2}\, \mathrm{d}x$

1. #### Transform fraction into power

$\int \frac{1}{x^2}\, \mathrm{d}x=\int x^{-2}\, \mathrm{d}x$
2. #### Apply power rule

$\int x^{-2}\, \mathrm{d}x=\frac{1}{-2+1}x^{-2+1}$ $=-x^{-1}$
3. #### Write power as fraction

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

### Note

Exception: When integrating $\frac{1}{x}=x^{-1}$, this rule does NOT apply, because then you can not use the power rule.

So you should remember this integral:
$\int \frac1x \,\mathrm{d}x=\ln|x|+C$

$\int 3\sqrt{x} \, \mathrm{d}x$

1. #### Transform root into power

(In this case, the constant factor rule is also used here)
$\int 3\sqrt{x} \, \mathrm{d}x=3\cdot \int x^\frac12\, \mathrm{d}x$
2. #### Apply power rule

$3\cdot \int x^\frac12 \, \mathrm{d}x=3\cdot\frac{1}{1,5}x^{\frac12+1}$ $=3\cdot\frac{2}{3}x^\frac32$
3. #### Rewrite power

$\int 3\sqrt{x} \, \mathrm{d}x=2x^\frac32$ $=2\sqrt{x^3}\color{purple}{+C}$