METHODS OF INTEGRATION – INTEGRATION BY SUBSTITUTION (Excercise Problems)

\[\underline{PART\ -\ A}\]
\[1.\ Evaluate\ :\ \int\ sec^2\ 5\ x\ dx\ \hspace{15cm}\]
\[\underline{PART\ -\ B}\]
\[2.\ Evaluate\ :\ \int\ cos^3\ 7\ x\ dx\ \hspace{15cm}\]
\[3.\ Evaluate\ :\ \int\ \frac{2x\ -\ 1}{{\sqrt{(x^2\ -\ x\ -\ 1)}}}\ dx\ \hspace{15cm}\]
\[4.\ Evaluate:\ \int\frac{sec^2\ x}{5\ +\ 4\ tan\ x\ dx}\ \hspace{15cm}\]
\[5.\ Evaluate:\ \int\frac{(tan^{-1}\ x)^3}{1\ +\ x^2}\ dx\ \hspace{15cm}\]
\[\underline{PART\ -\ C}\]
\[6.\ Evaluate:\ \hspace{2cm}\ (i)\ \int\ \frac{x\ +\ 1}{ x^2\ +\ 2x\ -\ 1}\ dx\ \hspace{2cm}\ (ii)\ \int\frac{sec^2\ x}{5\ +\ tan\ x}\ dx\ \hspace{10cm}\]
\[7.\ Evaluate:\ \hspace{2cm}\ (i)\ \int\ \frac{6x^2\ -\ 1}{ 2x^3\ -\ x\ +\ 5}\ dx\ \hspace{2cm}\ (ii)\ \int\frac{sin\ \sqrt{x}}{\sqrt{x}}\ dx\ \hspace{10cm}\]
\[8.\ Evaluate:\ \hspace{2cm}\ (i)\ \int\ \frac{cos\ x}{ (3\ -\ 5\ sin\ x)^6}\ dx\ \hspace{2cm}\ (ii)\ \int\frac{e^{tan^{-1}\ x}}{1\ +\ x^2}\ dx\ \hspace{10cm}\]
\[9.\ Evaluate:\ \hspace{2cm}\ (i)\ \int\ \frac{e^x}{1\ +\ e^x}\ dx\ \hspace{2cm}\ (ii)\ \int\ tan^5\ x\ sec^2\ x\ dx\ \hspace{10cm}\]