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# Integrals of Rational Functions Calculator

## Get detailed solutions to your math problems with our Integrals of Rational Functions step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

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###  Difficult Problems

1

Here, we show you a step-by-step solved example of integrals of rational functions. This solution was automatically generated by our smart calculator:

$\int\frac{2x^5-10x^3-2x^2+10}{x^2-5}$
2

Divide $2x^5-10x^3-2x^2+10$ by $x^2-5$

$\begin{array}{l}\phantom{\phantom{;}x^{2}-5;}{\phantom{;}2x^{3}\phantom{-;x^n}\phantom{-;x^n}-2\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-5\overline{\smash{)}\phantom{;}2x^{5}\phantom{-;x^n}-10x^{3}-2x^{2}\phantom{-;x^n}+10\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-5;}\underline{-2x^{5}\phantom{-;x^n}+10x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-2x^{5}+10x^{3};}-2x^{2}\phantom{-;x^n}+10\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-5-;x^n;}\underline{\phantom{;}2x^{2}\phantom{-;x^n}-10\phantom{;}\phantom{;}}\\\phantom{;\phantom{;}2x^{2}-10\phantom{;}\phantom{;}-;x^n;}\\\end{array}$
3

Resulting polynomial

$\int\left(2x^{3}-2\right)dx$
4

Expand the integral $\int\left(2x^{3}-2\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$\int2x^{3}dx+\int-2dx$

The integral of a function times a constant ($2$) is equal to the constant times the integral of the function

$2\int x^{3}dx$

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $3$

$\frac{1}{2}x^{4}$
5

The integral $\int2x^{3}dx$ results in: $\frac{1}{2}x^{4}$

$\frac{1}{2}x^{4}$

The integral of a constant is equal to the constant times the integral's variable

$-2x$
6

The integral $\int-2dx$ results in: $-2x$

$-2x$
7

Gather the results of all integrals

$\frac{1}{2}x^{4}-2x$
8

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{1}{2}x^{4}-2x+C_0$

##  Final answer to the problem

$\frac{1}{2}x^{4}-2x+C_0$

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