Integration by Parts Sqrt x ln x

In this tutorial we shall drive integral of Sqrt x lnx, and solve this problem with the help of integration by parts method.

The integral of Sqrt x lnx is of the form

\begin{gathered} I = \int {\sqrt x \ln xdx} \\ \Rightarrow I = \int {\ln x\sqrt x }  dx\,\,\,\,{\text{ -  -  - }}\left( {\text{i}} \right) \\ \end{gathered}

Here first function is \ln  x and second function will be \sqrt  x
Using formula for integration by parts, we have

\int  {\left[ {f\left( x \right)g\left( x \right)} \right]dx = f\left( x \right)\int  {g\left( x \right)dx - \int {\left[ {\frac{d}{{dx}}f\left( x \right)\int  {g\left( x \right)dx} } \right]dx} } }

Equation (i) becomes using above formula, we have

\begin{gathered} I = \ln x\int {\sqrt x dx - \int {\left[  {\frac{d}{{dx}}\ln x\left( {\int {\sqrt x dx} } \right)} \right]} dx} \\ \Rightarrow I = \ln  x\frac{{{x^{\frac{3}{2}}}}}{{\frac{3}{2}}} - \int {\left[  {\frac{1}{x}\frac{{{x^{\frac{3}{2}}}}}{{\frac{3}{2}}}} \right]} dx \\ \Rightarrow I =  \frac{2}{3}{x^{\frac{3}{2}}}\ln x - \frac{2}{3}\int {{x^{\frac{1}{2}}}} dx \\ \Rightarrow I =  \frac{2}{3}{x^{\frac{3}{2}}}\ln x -  \frac{2}{3}\frac{{{x^{\frac{3}{2}}}}}{{\frac{3}{2}}} + c \\ \Rightarrow I =  \frac{2}{3}{x^{\frac{3}{2}}}\ln x - \frac{4}{9}{x^{\frac{3}{2}}} + c \\ \Rightarrow \int {\ln x\sqrt x } dx =  \frac{2}{3}{x^{\frac{3}{2}}}\ln x - \frac{4}{9}{x^{\frac{3}{2}}} + c \\ \end{gathered}