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Evaluate the indefinite integral:
\(\mathsf{\displaystyle\int 11\,\ell n(^3\hspace{-5}\sqrt{x})\,dx}\\\\\\ \mathsf{\displaystyle\int\!\ell n(^3\hspace{-5}\sqrt{x})\cdot 11\,dx\qquad\quad(i)}\)
Integration by parts:
\(\begin{array}{lcl}
\mathsf{u=\,\ell n(^3\hspace{-5}\sqrt{x})}&\quad\Rightarrow\quad&\mathsf{du=\dfrac{1}{^3\hspace{-5}\sqrt{x}}\cdot \dfrac{d}{dx}(^3\hspace{-5}\sqrt{x})\,dx}\\\\
&&\mathsf{du=x^{-1/3}\cdot \dfrac{1}{3}\,x^{(1/3)-1}\,dx}\\\\
&&\mathsf{du=\dfrac{1}{3}\,x^{-1/3}\cdot x^{-2/3}\,dx}\\\\
&&\mathsf{du=\dfrac{1}{3}\,x^{-1}\,dx}
\\\\\\
\mathsf{dv=11\,dx}&\quad\Leftarrow\quad&\mathsf{v=11x}
\end{array}\)
\(\mathsf{\displaystyle\int\! u\,dv=u\cdot v-\int\!v\,du}\\\\\\
\mathsf{\displaystyle\int\!\ell n(^3\hspace{-5}\sqrt{x})\cdot 11\,dx=\ell n(^3\hspace{-5}\sqrt{x})\cdot 11x-\int\!11x\cdot \frac{1}{3}\,x^{-1}\,dx}\\\\\\
\boxed{\begin{array}{c}\mathsf{\displaystyle\int\!11\,\ell n(^3\hspace{-5}\sqrt{x})\,dx=11x\,\ell n(^3\hspace{-5}\sqrt{x})-\frac{11}{3}x+C}\end{array}}\quad\longleftarrow\quad\textsf{this is the answer.}\)
I hope this helps. =)
