SOLUTIONS TO INTERNAL ASSESSMENT – II FOR ENGINEERING MATHEMATICS-II

\[\underline{PART\ -\ A}\]
\[Marks:\ 6\ ×\ 1\ =\ 6\]
\[1.\ \color{Red}{Evaluate: \int(x^2 -x + 5)\ dx}\ \hspace{15cm}\]

Soln: Refer Example 1 under Integration Decomposition Method

\[2.\ \color{red}{Evaluate\ :\ \int\ \frac{2x}{1 + x^2}}\ \hspace{18cm}\]

Soln: Refer Example 3 under Integration By Substitution

\[3.\ \color{red}{Evaluate\ :\ \int\ \frac{dx}{x^2 + 4}}\ \hspace{18cm}\]

Soln: Refer Example No. 1 (STANDARD INTEGRALS)

\[4.\ \color{Red}{Evaluate:\ \int x\ sin\ x\ dx}\ \hspace{18cm}\]

Soln: Refer Example No. 1 INTEGRATION BY PARTS

\[5.\ \color{red}{Evaluate:\ \int x^3\ sin\ x\ dx}\ \hspace{18cm}\]

Soln: Refer Example No. 1 (BERNOULLI’S FORMULA)

\[6.\ \color{red}{Evaluate\ :\ \int_1^2 x^2\ dx}\ \hspace{18cm}\]

Soln: Refer Example No. 1 DEFINITE INTEGRALS

\[\underline{PART\ -\ B}\]
\[Marks:\ 7\ ×\ 2\ =\ 14\]
\[7.\ \color{red}{Evaluate: \int(x^2 + x + 1) ( x^2 – x + 1)\ dx}\ \hspace{55cm}\]

Soln: Refer Example 6 under Integration Decomposition Method

\[8.\ \color {red}{Evaluate: \int \frac{sin^2 x}{ 1+ cos x} \ dx}\ \hspace{15cm}\]

Soln: Refer Example 6 under Integration Decomposition Method (Excercise Problems with Solutions)

\[9.\ \color{red}{Evaluate\ :\ \int\ (x^2 + x + 1)^5\ (2x + 1) dx}\ \hspace{15cm}\]

Soln: Refer Example 3 under Integration By Substitution

\[10.\ \color{red}{Evaluate:\ \int \frac{dx}{4x^2 + 9}}\ \hspace{18cm}\]

Soln: Refer Example No. 2 (STANDARD INTEGRALS)

\[11.\ \color{red}{Evaluate:\ \int e^{nx}\ dx}\ \hspace{18cm}\]

Soln: Refer Example No. 3 INTEGRATION BY PARTS

\[12.\ \color{red}{\int x^3\ cos\ x\ dx}\ \hspace{18cm}\]

Soln: Refer Example No. 2 (BERNOULLI’S FORMULA)

\[13.\ \color{red}{Evaluate\ :\ \int_0^\frac{\pi}{2} sin^3 x\ dx}\ \hspace{18cm}\]

Soln: Refer Example No. 6 DEFINITE INTEGRALS

\[\underline{PART\ -\ C}\]
\[14.\ \color{red}{i)\ Evaluate:\ \hspace{2cm}\ (i)\ \int \frac{cos^2 x}{ 1- sin x} \ dx\ \hspace{2cm}\ (ii)\ \int sin 5x\ cos 2x\ dx}\ \hspace{10cm}\]

Soln: Refer Example No. 5 and Example No. 8 under Trigonometry Related Formulae (INTEGRATION DECOMPOSITON METHOD (TEXT)

\[\hspace{1cm}\ \color{red}{ii)\ Evaluate:\ \hspace{2cm}\ (i)\ \int\ \frac{sec^2(log x)}{ x}\ dx\ \hspace{2cm}\ (ii)\ \int \frac{dx}{x^2 – 36}}\ \hspace{10cm}\]

Soln: Refer Example 7 under Integration By Substitution

Refer Example No. 2 (STANDARD INTEGRALS)

\[15.\ \color{red}{i)\ Evaluate:\ \hspace{2cm}\ (i)\ \int x\ sin\ 2x\ dx\ \hspace{2cm}\ (ii)\ \int x^2\ sin\ 2x\ dx}\ \hspace{10cm}\]

Soln: Refer Example No. 5 INTEGRATION BY PARTS

Refer Example No. 2 (BERNOULLI’S FORMULA)

\[\hspace{1cm}\ ii.\ \color{red}{Evaluate: \int_0^\frac{\pi}{2} (2 + sin x)^3 cos x \ dx}\ \hspace{15cm}\]

Soln: Refer Example No. 9 DEFINITE INTEGRALS

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