# Power series Calculator

## Get detailed solutions to your math problems with our Power series 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

Solved example of power series

$\int\frac{\cos\left(x\right)}{x}dx$
2

Use the Taylor series for rewrite the function $\cos\left(x\right)$ as an approximation: $\displaystyle f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$, with $a=0$. Here we will use only the first four terms of the serie

$\int\frac{\frac{1}{0!}+x^{2}\frac{-1}{2!}+x^{4}\frac{1}{4!}+x^{6}\frac{-1}{6!}}{x}dx$
3

Split the fraction $\frac{1-\frac{1}{2}x^{2}+\frac{1}{24}x^{4}-\frac{1}{720}x^{6}}{x}$ inside the integral, in two terms with common denominator $x$

$\int\left(\frac{1}{x}+\frac{-\frac{1}{2}x^{2}+\frac{1}{24}x^{4}-\frac{1}{720}x^{6}}{x}\right)dx$

Split the fraction $\frac{-\frac{1}{2}x^{2}+\frac{1}{24}x^{4}-\frac{1}{720}x^{6}}{x}$ in two terms with common denominator $x$

$\int\left(\frac{1}{x}+\frac{-\frac{1}{2}x^{2}}{x}+\frac{\frac{1}{24}x^{4}-\frac{1}{720}x^{6}}{x}\right)dx$

Split the fraction $\frac{\frac{1}{24}x^{4}-\frac{1}{720}x^{6}}{x}$ in two terms with common denominator $x$

$\int\left(\frac{1}{x}+\frac{-\frac{1}{2}x^{2}}{x}+\frac{\frac{1}{24}x^{4}}{x}+\frac{-\frac{1}{720}x^{6}}{x}\right)dx$
4

Split the fraction $\frac{\frac{1}{24}x^{4}-\frac{1}{720}x^{6}}{x}$ in two terms with common denominator $x$

$\int\left(\frac{1}{x}+\frac{-\frac{1}{2}x^{2}}{x}+\frac{\frac{1}{24}x^{4}}{x}+\frac{-\frac{1}{720}x^{6}}{x}\right)dx$

Simplify the fraction by $x$

$\int\left(\frac{1}{x}-\frac{1}{2}x+\frac{1}{24}x^{3}-\frac{1}{720}x^{5}\right)dx$

The integral of the sum of two or more functions is equal to the sum of their integrals

$\int\frac{1}{x}dx+\int-\frac{1}{2}xdx+\int\frac{1}{24}x^{3}dx+\int-\frac{1}{720}x^{5}dx$
5

Simplifying

$\int\frac{1}{x}dx+\int-\frac{1}{2}xdx+\int\frac{1}{24}x^{3}dx+\int-\frac{1}{720}x^{5}dx$
6

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+\int\frac{1}{24}x^{3}dx+\int-\frac{1}{720}x^{5}dx$
7

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+\frac{1}{24}\int x^{3}dx+\int-\frac{1}{720}x^{5}dx$
8

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+\frac{1}{24}\int x^{3}dx-\frac{1}{720}\int x^{5}dx$

Add the values $3$ and $1$

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+\frac{1}{24}\frac{x^{4}}{4}-\frac{1}{720}\int x^{5}dx$

Multiplying the fraction and term

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+\frac{\frac{1}{24}x^{4}}{4}-\frac{1}{720}\int x^{5}dx$
9

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

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+\frac{\frac{1}{24}x^{4}}{4}-\frac{1}{720}\int x^{5}dx$

Take $\frac{\frac{1}{24}}{4}$ out of the fraction

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+x^{4}\frac{\frac{1}{24}}{4}-\frac{1}{720}\int x^{5}dx$

Divide $\frac{1}{24}$ by $4$

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+\frac{1}{96}x^{4}-\frac{1}{720}\int x^{5}dx$
10

Simplifying

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+\frac{1}{96}x^{4}-\frac{1}{720}\int x^{5}dx$

Add the values $5$ and $1$

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+\frac{1}{96}x^{4}-\frac{1}{720}\frac{x^{6}}{6}$

Multiplying the fraction and term

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+\frac{1}{96}x^{4}+\frac{-\frac{1}{720}x^{6}}{6}$
11

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

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+\frac{1}{96}x^{4}+\frac{-\frac{1}{720}x^{6}}{6}$

Take $\frac{-\frac{1}{720}}{6}$ out of the fraction

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+\frac{1}{96}x^{4}+x^{6}\frac{-\frac{1}{720}}{6}$

Divide $-\frac{1}{720}$ by $6$

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+\frac{1}{96}x^{4}-\frac{1}{4329}x^{6}$
12

Simplifying

$\int\frac{1}{x}dx-\frac{1}{2}\int xdx+\frac{1}{96}x^{4}-\frac{1}{4329}x^{6}$

Multiply $-\frac{1}{2}$ times $\frac{1}{2}$

$\int\frac{1}{x}dx-\frac{1}{4}x^2+\frac{1}{96}x^{4}-\frac{1}{4329}x^{6}$
13

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\int\frac{1}{x}dx-\frac{1}{4}x^2+\frac{1}{96}x^{4}-\frac{1}{4329}x^{6}$

Any expression multiplied by $1$ is equal to itself

$\ln\left|x\right|-\frac{1}{4}x^2+\frac{1}{96}x^{4}-\frac{1}{4329}x^{6}$
14

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

$\ln\left|x\right|-\frac{1}{4}x^2+\frac{1}{96}x^{4}-\frac{1}{4329}x^{6}$
15

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

$\ln\left|x\right|-\frac{1}{4}x^2+\frac{1}{96}x^{4}-\frac{1}{4329}x^{6}+C_0$

$\ln\left|x\right|-\frac{1}{4}x^2+\frac{1}{96}x^{4}-\frac{1}{4329}x^{6}+C_0$