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

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