What is the antiderivative of ln(x)/sqrtx ln(x)x?

1 Answer
Dec 26, 2015

2sqrt(x)(ln(x)-2) + C2x(ln(x)2)+C

Explanation:

For the given function, finding the antiderivative is equivalent to finding the indefinite integral. We will proceed by applying Integration by Parts.

Let u = ln(x)u=ln(x) and dv = 1/sqrt(x)dxdv=1xdx

Then du = 1/xdxdu=1xdx and v = 2sqrt(x)v=2x

From the integration by parts formula intudv = uv - intvduudv=uvvdu

intln(x)/sqrt(x)dx = 2sqrt(x)ln(x) - int2sqrt(x)*1/xdxln(x)xdx=2xln(x)2x1xdx

= 2sqrt(x)ln(x) - 2int1/sqrt(x)dx=2xln(x)21xdx

= 2sqrt(x)ln(x) - 2(2sqrt(x)) + C=2xln(x)2(2x)+C

= 2sqrt(x)(ln(x)-2) + C=2x(ln(x)2)+C