Notes: What is the antiderivative of the function lnx/square root x?

Thursday, April 9, 2015

To determine the antiderivative of the given function,
we'll have to calculate the indefinite integral of the given
function:

$\displaystyle\int$ ln x
dx/ $\displaystyle\sqrt{{{x}}}$

We'll integrate by parts using the
formula:

$\displaystyle\int$ udv = uv - $\displaystyle\int$
vdu

Let u = ln x => du =
dx/x

Let dv = dx/ $\displaystyle\sqrt{{{x}}}$ => v =
2 $\displaystyle\sqrt{{{x}}}$

$\displaystyle\int$ ln xdx/ $\displaystyle\sqrt{{{x}}}$ = 2ln x* $\displaystyle\sqrt{{{x}}}$ - $\displaystyle\int$
2 $\displaystyle\sqrt{{{x}}}$ dx/x

$\displaystyle\int$ ln xdx/ $\displaystyle\sqrt{{{x}}}$ =
ln(x^2)* $\displaystyle\sqrt{{{x}}}$ - 2x^(-1/2 + 1)/(1 - 1/2) + C

$\displaystyle\int$ ln x
dx/ $\displaystyle\sqrt{{{x}}}$ = ln(x^2)* $\displaystyle\sqrt{{{x}}}$ - 4 $\displaystyle\sqrt{{{x}}}$ + C

$\displaystyle\int$ ln
x dx/ $\displaystyle\sqrt{{{x}}}$ = $\displaystyle\sqrt{{{x}}}$ [ln (x^2) - 4] +
C

The antiderivative of the function is Y =
$\displaystyle\sqrt{{{x}}}$ [ln (x^2) - 4] + C

Notes