What is the antiderivative of #lnsqrt( 5 x )#?

1 Answer
Lovecraft
Jan 9, 2016

#I = (x(ln(5)+ln(x) - 1))/2 + c#

Explanation:

#I = intln(sqrt(5x))dx#

Using log properties

#I = 1/2int(ln(5) + ln(x))dx#

#I = (xln(5))/2 + 1/2intln(x)dx#

For the last integral say #u = ln(x)# so #du = 1/x# and #dv = 1# so #v = x#

#I = (xln(5))/2 + 1/2(ln(x)x - intx/xdx)#
#I = (x(ln(5)+ln(x) - 1))/2 + c#

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