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Step-by-step Solution

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

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ln
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asin
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Answer

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

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 ($x$)

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

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Answer

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

Main topic:

Integral calculus

Used formulas:

4. See formulas

Time to solve it:

~ 1.37 seconds

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