BERNOULLI’S FORMULA(Excercise Problems with Solutions)

\[\underline{PART\ -\ A}\]
\[1.\ \color {red}{Evaluate\ :\ \int x^2\ sin\ x\ dx}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[ ILATE \]
\[u= x^2\ \hspace{2cm}\ dv = sin\ x\ dx\]
\[v = – cos x\]
\[u^! = 2x\ \hspace{2cm}\ v_1 = – sin\ x\]
\[u^{!!} = 2\ \hspace{2cm}\ v_2 = cos \ x\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 \ +\ c\]
\[\int x^2\ sin\ x\ dx = x^2(-cos x)\ -\ 2x (- sin\ x)\ +\ 2 (cos\ x)\ + c\]
\[\boxed{\int x^2\ sin\ x\ dx = x^2(-cos x)\ -\ 2x (- sin\ x)\ +\ 2 (cos\ x)+c}\]
\[2.\ \color {red}{Evaluate\ :\ \int x^2\ e^x\ dx}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[ ILATE \]
\[u= x^2\ \hspace{2cm}\ dv = e^x\ dx\]
\[u^! = 2x\ \hspace{2cm}\ v\ =\ e^x \]
\[u^{!!} = 2\ \hspace{2cm}\ v_1 = e^x\]
\[\hspace{3cm}\ v_2 = e^x\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2\]
\[\int x^2\ e^x\ dx = x^2\ e^x\ – 2x\ e^x\ +\ 2\ e^x\ + c\]
\[\boxed{\int x^2\ e^x\ dx = x^2\ e^x\ – 2x\ e^x\ +\ 2\ e^x+c}\]
\[\underline{PART\ -\ B}\]
\[3.\ \color {red}{Evaluate\ :\ \int x^2\ cos\ 3x\ dx}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[ ILATE \]
\[u= x^2\ \hspace{2cm}\ dv = cos\ 3x\ dx\]
\[u^! = 2x\ \hspace{2cm}\ v = \frac{sin\ 3x}{3}\]
\[u^{!!} = 2\ \hspace{2cm}\ v_1 = – \frac{cos\ 3x}{9}\]
\[\hspace{3cm}\ v_2 = -\frac{sin\ 3x}{27}\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 \]
\[\int x^2\ cos\ 3x\ dx = x^2(\frac{sin\ 3x}{3}) – 2x (-\frac{cos\ 3x}{9}) + 2(- \frac{sin\ 3x}{27}) + c\]
\[\boxed{\int x^2\ cos\ 3x\ dx = x^2\frac{sin\ 3x}{3}\ +\ 2x \frac{cos\ 3x}{9}\ -\ 2\frac{sin\ 3x}{27}+c}\]
\[\underline{PART\ -\ C}\]
\[4.\ \color{red}{Evaluate:\ \hspace{2cm}\ (i)\ \int\ x^2\ sin\ 4x\ dx\ \hspace{2cm}\ (ii)\ \int x^2\ e^{-7x}\ dx}\ \hspace{10cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[(i)\ \int\ x^2\ sin\ 4x\ dx\ \hspace{10cm}\]
\[ ILATE \]
\[u= x^2\ \hspace{2cm}\ dv = sin\ 4x\ dx\]
\[u^! = 2x\ \hspace{2cm}\ v = – \frac{cos\ 4x}{4}\]
\[u^{!!} = 2\ \hspace{2cm}\ v_1 = – \frac{sin\ 4x}{16}\]
\[\hspace{3cm}\ v_2 = \frac{cos\ 4x}{64}\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 \]
\[\int x^2\ sin\ 4x\ dx = x^2(-\frac{cos\ 4x}{4}) – 2x (-\frac{sin\ 4x}{16}) + 2(\frac{cos\ 4x}{16}) + c\]
\[ = – x^2\frac{cos\ 4x}{4} + x \frac{sin\ 4x}{8} + \frac{cos\ 4x}{8} + c\]
\[\boxed{\int x^2\ sin\ 4x\ dx = – x^2\frac{cos\ 4x}{4} + x \frac{sin\ 4x}{8} + \frac{cos\ 4x}{8}+c}\]
\[(ii)\ \int\ x^2\ e^{-7x}\ dx\ \hspace{10cm}\]
\[ ILATE \]
\[u= x^2\ \hspace{2cm}\ dv = e^{-7x}\ dx\]
\[u^! = 2x\ \hspace{2cm}\ v = \frac{e^{-7x}}{-7}\]
\[u^{!!} = 2\ \hspace{2cm}\ v_1 = \frac{e^{-7x}}{49}\]
\[\hspace{3cm}\ v_2 = \frac{e^{-7x}}{-343}\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 \]
\[\int x^2\ e^{-7x}\ dx = x^2\frac{e^{-7x}}{-7}\ -\ 2\ x \frac{e^{-7x}}{49}\ +\ 2\frac{e^{-7x}}{-343}\ +\ c\]
\[ = -\ x^2\frac{e^{-7x}}{7}\ -\ 2\ x \frac{e^{-7x}}{49}\ -\ 2\frac{e^{-7x}}{343} + c\]
\[\boxed{\int x^2\ e^{-7x}\ dx = -\ x^2\frac{e^{-7x}}{7}\ -\ 2\ x \frac{e^{-7x}}{49}\ -\ 2\frac{e^{-7x}}{343}+c}\]
\[5.\ \color{red}{Evaluate:\ \hspace{2cm}\ (i)\ \int\ x^2\ cos\ 5x\ dx\ \hspace{2cm}\ (ii)\ \int x^2\ e^{-2x}\ dx}\ \hspace{10cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[(i)\ \int\ x^2\ cos\ 5x\ dx\ \hspace{10cm}\]
\[ ILATE \]
\[u= x^2\ \hspace{2cm}\ dv = cos\ 5x\ dx\]
\[u^! = 2x\ \hspace{2cm}\ v = \frac{sin\ 5x}{5}\]
\[u^{!!} = 2\ \hspace{2cm}\ v_1 = – \frac{cos\ 5x}{25}\]
\[\hspace{3cm}\ v_2 = -\frac{sin\ 5x}{125}\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 \]
\[\int x^2\ cos\ 5x\ dx = x^2(\frac{sin\ 5x}{5}) – 2x (-\frac{cos\ 5x}{25}) + 2(- \frac{sin\ 5x}{125}) + c\]
\[ = x^2\frac{sin\ 2x}{2}\ +\ 2x \frac{cos\ 5x}{25} – 2\frac{sin\ 5x}{125} + c\]
\[\boxed{\int x^2\ cos\ 5x\ dx = x^2\frac{sin\ 2x}{2}\ +\ 2x \frac{cos\ 5x}{25} – 2\frac{sin\ 5x}{125}+c}\]
\[(ii)\ \int\ x^2\ e^{-2x}\ dx\ \hspace{10cm}\]
\[ ILATE \]
\[u= x^2\ \hspace{2cm}\ dv = e^{-2x}\ dx\]
\[u^! = 2x\ \hspace{2cm}\ v = \frac{e^{-2x}}{-2}\]
\[u^{!!} = 2\ \hspace{2cm}\ v_1 = \frac{e^{-2x}}{4}\]
\[\hspace{3cm}\ v_2 = \frac{e^{-2x}}{-8}\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 \]
\[\int x^2\ e^{-2x}\ dx = x^2\frac{e^{-2x}}{-2}\ -\ 2\ x \frac{e^{-2x}}{4}\ +\ 2\frac{e^{-2x}}{-8}\ +\ c\]
\[ = -\ x^2\frac{e^{-2x}}{2}\ -\ x \frac{e^{-2x}}{2}\ -\ \frac{e^{-2x}}{4} + c\]
\[\boxed{\int x^2\ e^{-2x}\ dx = -\ x^2\frac{e^{-2x}}{2}\ -\ x \frac{e^{-2x}}{2}\ -\ \frac{e^{-2x}}{4}+c}\]
\[6.\ \color{red}{Evaluate:\ \int x^2\ cos\ 6x\ dx}\ \hspace{18cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[ ILATE \]
\[u= x^2\ \hspace{2cm}\ dv = cos\ 6x\ dx\]
\[u^! = 2x\ \hspace{2cm}\ v = \frac{sin\ 6x}{6}\]
\[u^{!!} = 2\ \hspace{2cm}\ v_1 = – \frac{cos\ 6x}{36}\]
\[\hspace{3cm}\ v_2 = -\frac{sin\ 6x}{216}\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 \]
\[\int x^2\ cos\ 6x\ dx = x^2(\frac{sin\ 6x}{6}) – 2x (-\frac{cos\ 6x}{36}) + 2(- \frac{sin\ 6x}{216}) + c\]
\[ = x^2\frac{sin\ 6x}{6} + x \frac{cos\ 6x}{18} – \frac{sin\ 6x}{108} + c\]
\[\boxed{\int x^2\ cos\ 6x\ dx = x^2\frac{sin\ 6x}{6} + x \frac{cos\ 6x}{18} – \frac{sin\ 6x}{108}+c}\]
\[7.\ \color{red}{Evaluate:\ \hspace{2cm}\ (i)\ \int\ x^2\ e^{-3x}\ dx\ \hspace{2cm}\ (ii)\ \int x^2\ sin\ 6x\ dx}\ \hspace{10cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[(i)\ \int\ x^2\ e^{-3x}\ dx\ \hspace{10cm}\]
\[ ILATE \]
\[u= x^2\ \hspace{2cm}\ dv = e^{-3x}\ dx\]
\[u^! = 2x\ \hspace{2cm}\ v = \frac{e^{-3x}}{-3}\]
\[u^{!!} = 2\ \hspace{2cm}\ v_1 = \frac{e^{-3x}}{9}\]
\[\hspace{3cm}\ v_2 = \frac{e^{-3x}}{-27}\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 \]
\[\int x^2\ e^{-3x}\ dx = x^2\frac{e^{-3x}}{-3}\ -\ 2\ x \frac{e^{-3x}}{9}\ +\ 2\frac{e^{-3x}}{-27}\ +\ c\]
\[ = -\ x^2\frac{e^{-3x}}{3}\ -\ 2x \frac{e^{-3x}}{9}\ -\ 2\frac{e^{-3x}}{27} + c\]
\[\boxed{\int x^2\ e^{-3x}\ dx = -\ x^2\frac{e^{-3x}}{3}\ -\ 2x \frac{e^{-3x}}{9}\ -\ 2\frac{e^{-3x}}{27}+c}\]
\[(ii)\ \int\ x^2\ sin\ 6x\ dx\ \hspace{10cm}\]
\[ ILATE \]
\[u= x^2\ \hspace{2cm}\ dv = sin\ 6x\ dx\]
\[u^! = 2x\ \hspace{2cm}\ v = – \frac{cos\ 6x}{6}\]
\[u^{!!} = 2\ \hspace{2cm}\ v_1 = – \frac{sin\ 6x}{36}\]
\[\hspace{3cm}\ v_2 = \frac{cos\ 6x}{216}\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 \]
\[\int x^2\ sin\ 6x\ dx = x^2(-\frac{cos\ 6x}{6}) – 2x (-\frac{sin\ 6x}{36}) + 2(\frac{cos\ 6x}{216}) + c\]
\[ = – x^2\frac{cos\ 6x}{6} + x \frac{sin\ 6x}{18} + \frac{cos\ 6x}{108} + c\]
\[\boxed{\int x^2\ sin\ 6x\ dx = – x^2\frac{cos\ 6x}{6} + x \frac{sin\ 6x}{18} + \frac{cos\ 6x}{108}+c}\]
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