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How do you find the anti derivative of #(sqrtx -8)/(sqrtx-4)#?

Answer 1

Ezekiel Alexander

#int (sqrtx -8)/(sqrtx -4) dx=x-8sqrtx -32ln|sqrtx -4|# +C

The expression is equivalent to# 1 -4/(sqrtx -4)#

#int (sqrtx -8)/(sqrtx -4) dx#= #int (1 -4/(sqrtx -4) )dx= x -4int 1/(sqrtx -4) dx#

Now to integrate # 1/(sqrtx -4) dx#, let x =t^2, so that dx= 2tdt and so #int 1/(sqrtx-4) dx= int (2tdt)/(t-4)#

=#2int 1+4/(t-4) dt#= 2(t+4 ln|t-4))=#2sqrtx +8ln|sqrtx-4|#

#int (sqrtx -8)/(sqrtx -4) dx=x-8sqrtx -32ln|sqrtx -4|# +C

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Answer 2

Evelyn Agard

To find the antiderivative of (\frac{\sqrt{x} - 8}{\sqrt{x} - 4}), you can perform polynomial long division or use partial fraction decomposition to express the fraction in a form that is easier to integrate. Once the fraction is decomposed, integrate each term separately.

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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.