# Step-by-step Solution

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## Step-by-step explanation

Problem to solve:

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

Learn how to solve calculus problems step by step online.

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

Learn how to solve calculus problems step by step online. Integrate int(((cos(x)/x))dx with respect to x. 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. Split the fraction \frac{1+\frac{-x^{2}}{2}+\frac{1}{24}x^{4}-\frac{1}{720}x^{6}}{x} inside the integral, in two terms with common denominator x. The integral of the sum of two or more functions is equal to the sum of their integrals. The integral \int\frac{1}{x}dx results in: \ln\left|x\right|.

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

### Problem Analysis

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

Calculus

~ 2.0 seconds