What is the antiderivative of lnsqrt( 5 x )ln√5x?

1 Answer
Lovecraft
Jan 9, 2016

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

Explanation:

I = intln(sqrt(5x))dxI=∫ln(√5x)dx

Using log properties

I = 1/2int(ln(5) + ln(x))dxI=12∫(ln(5)+ln(x))dx

I = (xln(5))/2 + 1/2intln(x)dxI=xln(5)2+12∫ln(x)dx

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

I = (xln(5))/2 + 1/2(ln(x)x - intx/xdx)I=xln(5)2+12(ln(x)x−∫xxdx)
I = (x(ln(5)+ln(x) - 1))/2 + cI=x(ln(5)+ln(x)−1)2+c

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