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What is the antiderivative of #ln x / x^(1/2)#?

Answer 1

William Abdalla

#int lnx/x^(1/2) dx = 2 sqrtx(lnx - 2)+C#

Combine in segments:

#int lnx/x^(1/2) dx = int lnx * x^(-1/2)dx#

#int lnx/x^(1/2) dx = 2 int lnx * d/dx (x^(1/2)) dx#

#int lnx/x^(1/2) dx = 2 x^(1/2)lnx - 2 int x^(1/2)*d/dx(lnx)dx#

#int lnx/x^(1/2) dx = 2 x^(1/2)lnx - 2 int x^(1/2)*1/xdx#

#int lnx/x^(1/2) dx = 2 x^(1/2)lnx - 2 int x^(-1/2)dx#

#int lnx/x^(1/2) dx = 2 x^(1/2)lnx - 4 x^(+1/2)+C#

#int lnx/x^(1/2) dx = 2 x^(1/2)(lnx - 2)+C#

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Answer 2

Emma Aldridge

The antiderivative of ln(x) / x^(1/2) is 2√x ln(x) - 4√x + C, where C is the constant of integration.

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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.