# Step-by-step Solution

## Integrate (cos(x)/x

Go
1
2
3
4
5
6
7
8
9
0
x
y
(◻)
◻/◻
2

e
π
ln
log
lim
d/dx
d/dx
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

### Videos

$-\frac{1}{4320}x^{6}+\ln\left|x\right|-\frac{1}{4}x^2+\frac{1}{96}x^{4}+C_0$

## Step-by-step explanation

Problem to solve:

$\int\left(\frac{cos\left(x\right)}{x}\right)dx$
1

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

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

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

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

$-\frac{1}{4320}x^{6}+\ln\left|x\right|-\frac{1}{4}x^2+\frac{1}{96}x^{4}+C_0$

### Struggling with math?

Access detailed step by step solutions to millions of problems, growing every day!

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

Taylor series

~ 1.22 seconds