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.

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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:
$\sqrt[n]{a^m}=a^{\frac{m}{n}}$
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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

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

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