How can we evaluate the indefinite integral of cos(x) / x dx using an infinite series?

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To evaluate the indefinite integral ∫ (cos(x) / x) dx using an infinite series, we can leverage the Taylor series expansion of the cosine function.

The Taylor series expansion for cos(x) around 0 is given by:

cos(x) = 1 - rac{x^2}{2!} + rac{x^4}{4!} - rac{x^6}{6!} + rac{x^8}{8!} - rac{x^{10}}{10!} + 	ext{...}

Substituting this series into the integral, we get:

∫ (cos(x) / x) dx = ∫ (1/x) igg(1 - rac{x^2}{2!} + rac{x^4}{4!} - rac{x^6}{6!} + ...igg) dx

This simplifies to:

∫ igg( rac{1}{x} - rac{x}{2!} + rac{x^3}{4!} - rac{x^5}{6!} + ... igg) dx

Now we can integrate each term separately:

  • For ∫ (1/x) dx, the result is ln|x| + C.
  • For ∫ (- rac{x}{2!}) dx, the result is - rac{x^2}{2*2!} = - rac{x^2}{4} + C.
  • For ∫ ( rac{x^3}{4!}) dx, the result is rac{x^4}{4*4!} = rac{x^4}{96} + C.
  • Continuing this process, we will evaluate each term of the series.

After integrating term by term, we can represent the indefinite integral as:

∫ (cos(x) / x) dx = ln|x| - rac{x^2}{4} + rac{x^4}{96} - rac{x^6}{720} + ... + C

This expression can be written as:

∫ (cos(x) / x) dx = ln|x| + 	ext{the sum of the series terms} + C

Thus, we arrive at the infinite series representation of the indefinite integral of cos(x) / x. Note that the sum diverges and does not converge to a closed-form expression, so it’s often presented in this series format.

In conclusion, using the series expansion of cosine allows us to evaluate the integral of cos(x) / x in a form that’s both useful and enlightening, highlighting the relationship between calculus and infinite series.

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