Answer
$$\frac{2}{9}{x^{3/2}}\left( {3\ln x - 2} \right) + C $$
Work Step by Step
$$\eqalign{
& \int {\sqrt x \ln x} dx \cr
& = \int {{x^{1/2}}\ln x} dx \cr
& {\text{Using integration by parts method }} \cr
& \,\,\,\,\,{\text{Let }}u = \ln x,\,\,\,\,du = \frac{1}{x}dx,\,\,\,\,dv = {x^{1/2}}dt,\,\,\,\,v = \frac{2}{3}{x^{3/2}} \cr
& {\text{Integration by parts then gives}} \cr
& \int {{x^{1/2}}\ln x} dx = \frac{2}{3}{x^{3/2}}\ln x - \int {\left( {\frac{2}{3}{x^{3/2}}} \right)\left( {\frac{1}{x}dx} \right)} \cr
& \int {{x^{1/2}}\ln x} dx = \frac{2}{3}{x^{3/2}}\ln x - \frac{2}{3}\int {{x^{1/2}}} dx \cr
& \int {{x^{1/2}}\ln x} dx = \frac{2}{3}{x^{3/2}}\ln x - \frac{2}{3}\left( {\frac{{{x^{3/2}}}}{{3/2}}} \right) + C \cr
& {\text{simplifying}} \cr
& \int {{x^{1/2}}\ln x} dx = \frac{2}{3}{x^{3/2}}\ln x - \frac{4}{9}{x^{3/2}} + C \cr
& \int {{x^{1/2}}\ln x} dx = \frac{2}{9}{x^{3/2}}\left( {3\ln x - 2} \right) + C \cr} $$
