INTEGRATION BY PARTS (Excercise Problems)

\[\underline{PART\ -\ B}\]
\[1.\ Evaluate\ :\ \int x\ e^{-2x}\ dx\ \hspace{15cm}\]
\[2.\ Evaluate\ :\ \int x\ sec^2 x\ dx\ \hspace{15cm}\]
\[\underline{PART\ -\ C}\]
\[3.\ Evaluate:\ \hspace{2cm}\ (i)\ \int\ x\ sin\ 5x\ dx\ \hspace{2cm}\ (ii)\ \int x\ log\ x\ dx\ \hspace{10cm}\]
\[4.\ Evaluate:\ \int x\ cos\ 3x\ dx\ \hspace{15cm}\]
\[5.\ Evaluate:\ \int x\ e^{4x}\ dx\ \hspace{15cm}\]
\[6.\ Evaluate:\ \hspace{2cm}\ (i)\ \int\ x^3\ log\ x\ dx\ \hspace{2cm}\ (ii)\ \int x\ e^{-5x}\ dx\ \hspace{10cm}\]