Integration by Parts Sqrt x ln x

In this tutorial we shall derive the integral of Sqrt x lnx, and solve this problem with the help of the 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 the first function is \ln x and the second function is \sqrt x

Using the 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} } }

Using the above formula, equation (i) becomes

\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}