What is the antiderivative of $ln \sqrt{5 x}$ $lnsqrt( 5 x )$ ?

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What is the antiderivative of $ln \sqrt{5 x}$ $lnsqrt( 5 x )$ ?

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Answer 1 · 2016-01-09T01:40:36

$I = \frac{x (ln (5) + ln (x) - 1)}{2} + c$ $I = (x(ln(5)+ln(x) - 1))/2 + c$

Explanation:

$I = \int ln (\sqrt{5 x}) d x$ $I = intln(sqrt(5x))dx$

Using log properties

$I = \frac{1}{2} \int (ln (5) + ln (x)) d x$ $I = 1/2int(ln(5) + ln(x))dx$

$I = \frac{x ln (5)}{2} + \frac{1}{2} \int ln (x) d x$ $I = (xln(5))/2 + 1/2intln(x)dx$

For the last integral say $u = ln (x)$ $u = ln(x)$ so $d u = \frac{1}{x}$ $du = 1/x$ and $d v = 1$ $dv = 1$ so $v = x$ $v = x$

$I = \frac{x ln (5)}{2} + \frac{1}{2} (ln (x) x - \int \frac{x}{x} d x)$ $I = (xln(5))/2 + 1/2(ln(x)x - intx/xdx)$
$I = \frac{x (ln (5) + ln (x) - 1)}{2} + c$ $I = (x(ln(5)+ln(x) - 1))/2 + c$

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What is the antiderivative of $ln \sqrt{5 x}$ $lnsqrt( 5 x )$ ?

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What is the antiderivative of $ln \sqrt{5 x}$ $lnsqrt( 5 x )$ ?