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![displaystyle intlimits { sqrt{x} ln x} , dx =bigg{u=ln x;dv = sqrt{x} dx;,,,, du= frac{dx}{x} ;,,, v= frac{2x^{ frac{3}{2} }}{3} bigg}=\\ \ = frac{2}{3} x^{3/2}ln x- frac{2}{3} intlimits { sqrt{x} } , dx = frac{2}{3} x^{3/2}ln x- frac{4x^{3/2}}{9} +C=\ \ \ =boxed{ frac{2}{9}x^{3/2}(3ln x-2)+C } displaystyle intlimits { sqrt{x} ln x} , dx =bigg{u=ln x;dv = sqrt{x} dx;,,,, du= frac{dx}{x} ;,,, v= frac{2x^{ frac{3}{2} }}{3} bigg}=\\ \ = frac{2}{3} x^{3/2}ln x- frac{2}{3} intlimits { sqrt{x} } , dx = frac{2}{3} x^{3/2}ln x- frac{4x^{3/2}}{9} +C=\ \ \ =boxed{ frac{2}{9}x^{3/2}(3ln x-2)+C }](https://tex.z-dn.net/?f=displaystyle++intlimits+%7B+sqrt%7Bx%7D+ln+x%7D+%2C+dx+%3Dbigg%7Bu%3Dln+x%3Bdv+%3D+sqrt%7Bx%7D+dx%3B%2C%2C%2C%2C+du%3D+frac%7Bdx%7D%7Bx%7D+%3B%2C%2C%2C+v%3D+frac%7B2x%5E%7B+frac%7B3%7D%7B2%7D+%7D%7D%7B3%7D+bigg%7D%3D%5C%5C+%5C+%3D+frac%7B2%7D%7B3%7D+x%5E%7B3%2F2%7Dln+x-+frac%7B2%7D%7B3%7D+intlimits+%7B+sqrt%7Bx%7D+%7D+%2C+dx+%3D+frac%7B2%7D%7B3%7D+x%5E%7B3%2F2%7Dln+x-+frac%7B4x%5E%7B3%2F2%7D%7D%7B9%7D+%2BC%3D%5C+%5C+%5C+%3Dboxed%7B+frac%7B2%7D%7B9%7Dx%5E%7B3%2F2%7D%283ln+x-2%29%2BC+%7D)
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