# Taylor series Calculator

## Get detailed solutions to your math problems with our Taylor 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

$\int\frac{\cos\left(x\right)}{x}dx$
3

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

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

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

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

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

$\ln\left|x\right|$
5

The integral $\int\frac{1}{x}dx$ results in: $\ln\left|x\right|$

$\ln\left|x\right|$

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

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

Simplifying

$\int-\frac{1}{2}xdx$

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

$-\frac{1}{2}\int xdx$

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

$-\frac{1}{4}x^2$
6

The integral $\int\frac{\frac{-x^{2}}{2}+\frac{x^{4}}{24}+\frac{-x^{6}}{720}}{x}dx$ results in: $-\frac{1}{4}x^2$

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

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

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

Simplifying

$\int\frac{x^{3}}{24}dx$

Take the constant out of the integral

$\frac{1}{24}\int x^{3}dx$

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

$\frac{1}{96}x^{4}$
7

The integral $\int\frac{\frac{x^{4}}{24}+\frac{-x^{6}}{720}}{x}dx$ results in: $\frac{1}{96}x^{4}$

$\frac{1}{96}x^{4}$

Simplify the fraction by $x$

$\int-\frac{1}{720}x^{5}dx$

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

$-\frac{1}{720}\int x^{5}dx$

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

$-\frac{1}{4329}x^{6}$
8

The integral $\int\frac{-\frac{1}{720}x^{6}}{x}dx$ results in: $-\frac{1}{4329}x^{6}$

$-\frac{1}{4329}x^{6}$
9

Gather the results of all integrals

$\ln\left|x\right|-\frac{1}{4}x^2+\frac{1}{96}x^{4}-\frac{1}{4329}x^{6}$
10

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

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

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