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What is the integral of #int (cosx)/(x) dx#?

Answer 1

This is one of those integrals that can't be done in terms of elementary functions. You can do it in terms of infinite series; and you can use various numerical methods to do the definite integral.

The Taylor series expansion of #cos(x)# is

#cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ...#

Dividing this by x gives us an infinite series expansion for #cos(x)/x#:

#cos(x)/x = 1/x - x/2! + (x^3)/4! - (x^5)/6! + ...#

And finally, integrating this series term by term gives us a power series expansion for the integral of cos(x)/x:

# int cos(x)/x dx = Ln(x) - (x^2)/(2*2!) + (x^4)/(4*4!) - (x^6)/(6*6!) + ... + c#

where #c# is the integration constant.

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Answer 2

Samantha Adkison

The integral of ( \int \frac{\cos(x)}{x} , dx ) does not have an elementary antiderivative expressible in terms of elementary functions. It is a well-known example of a non-elementary integral, often denoted by ( \text{Si}(x) ), where ( \text{Si}(x) ) represents the sine integral function. It is commonly computed numerically or expressed using special functions.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.