Respuesta :
The integration of the given integral ∫ [(cos x - 1)/x] dx will be given below.
[tex]\int \dfrac{\cos x -1}{x} \ dx = \Sigma _{n=0}^{\infty} (-1) ^n \ \dfrac{x^{2n}}{2n} \cdot \dfrac{1}{(2n)!} + C[/tex]
What is integration?
Integration is a way of finding the total by adding or summing the components. It's a reversal of differentiation, in which we break down functions into pieces. This approach is used to calculate the total on a large scale.
We know that
cos x = 1 - x²/2! + x⁴/4! - ... + (-1)ⁿ x²ⁿ/(2n)! + ....
The expansion series of the cos x can be written as
[tex]\rm \cos x = \Sigma _{n=0}^{\infty} (-1) ^n \ \dfrac{x^{2n}}{(2n)!}[/tex]
Then the integral will be given as
[tex]\rm \int \dfrac{\cos x -1}{x} \ dx = \Sigma _{n=0}^{\infty} (-1) ^n \ \dfrac{x^{2n}}{(2n)!} \cdot \dfrac{x^{2n-1}}{(2n)!}\\\\\\\int \dfrac{\cos x -1}{x} \ dx = \Sigma _{n=0}^{\infty} (-1) ^n \ \dfrac{x^{2n}}{2n} \cdot \dfrac{1}{(2n)!} + C[/tex]
The integration of the given integral ∫ [(cos x - 1)/x] dx will be given below.
[tex]\int \dfrac{\cos x -1}{x} \ dx = \Sigma _{n=0}^{\infty} (-1) ^n \ \dfrac{x^{2n}}{2n} \cdot \dfrac{1}{(2n)!} + C[/tex]
Where C is the constant of integration.
More about the integration link is given below.
https://brainly.com/question/18651211
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