# SOLUTIONS TO INTERNAL ASSESSMENT – II FOR ENGINEERING MATHEMATICS-II

$\underline{PART\ -\ A}$
$Marks:\ 6\ ×\ 1\ =\ 6$
$1.\ \color{Red}{Evaluate: \int(x^2 -x + 5)\ dx}\ \hspace{15cm}$
$2.\ \color{red}{Evaluate\ :\ \int\ \frac{2x}{1 + x^2}}\ \hspace{18cm}$
$3.\ \color{red}{Evaluate\ :\ \int\ \frac{dx}{x^2 + 4}}\ \hspace{18cm}$
$4.\ \color{Red}{Evaluate:\ \int x\ sin\ x\ dx}\ \hspace{18cm}$
$5.\ \color{red}{Evaluate:\ \int x^3\ sin\ x\ dx}\ \hspace{18cm}$
$6.\ \color{red}{Evaluate\ :\ \int_1^2 x^2\ dx}\ \hspace{18cm}$
$\underline{PART\ -\ B}$
$Marks:\ 7\ ×\ 2\ =\ 14$
$7.\ \color{red}{Evaluate: \int(x^2 + x + 1) ( x^2 – x + 1)\ dx}\ \hspace{55cm}$
$8.\ \color {red}{Evaluate: \int \frac{sin^2 x}{ 1+ cos x} \ dx}\ \hspace{15cm}$
$9.\ \color{red}{Evaluate\ :\ \int\ (x^2 + x + 1)^5\ (2x + 1) dx}\ \hspace{15cm}$

Soln: Refer Example 3 under Integration By Substitution

$10.\ \color{red}{Evaluate:\ \int \frac{dx}{4x^2 + 9}}\ \hspace{18cm}$
$11.\ \color{red}{Evaluate:\ \int e^{nx}\ dx}\ \hspace{18cm}$
$12.\ \color{red}{\int x^3\ cos\ x\ dx}\ \hspace{18cm}$
$13.\ \color{red}{Evaluate\ :\ \int_0^\frac{\pi}{2} sin^3 x\ dx}\ \hspace{18cm}$
$\underline{PART\ -\ C}$
$14.\ \color{red}{i)\ Evaluate:\ \hspace{2cm}\ (i)\ \int \frac{cos^2 x}{ 1- sin x} \ dx\ \hspace{2cm}\ (ii)\ \int sin 5x\ cos 2x\ dx}\ \hspace{10cm}$
$\hspace{1cm}\ \color{red}{ii)\ Evaluate:\ \hspace{2cm}\ (i)\ \int\ \frac{sec^2(log x)}{ x}\ dx\ \hspace{2cm}\ (ii)\ \int \frac{dx}{x^2 – 36}}\ \hspace{10cm}$
$15.\ \color{red}{i)\ Evaluate:\ \hspace{2cm}\ (i)\ \int x\ sin\ 2x\ dx\ \hspace{2cm}\ (ii)\ \int x^2\ sin\ 2x\ dx}\ \hspace{10cm}$
$\hspace{1cm}\ ii.\ \color{red}{Evaluate: \int_0^\frac{\pi}{2} (2 + sin x)^3 cos x \ dx}\ \hspace{15cm}$