BERNOULLI’S FORMULA

\[Integrals\ of\ the\ form\ \int x^m\ sin\ nx\ dx,\ \int x^m\ cos\ nx\ dx\ and\ \int x^m\ e^{nx}\ dx\ where\ m\le3\]

If u and v are functions x, then Bernoulli’s form of integration by parts formula is

\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 – u^{!!!}v_3 + …………\]

Where u΄, u΄΄,u΄΄΄….. are successive differentiation of the function u and v, v1, v2, v3, …………. the successive integration of the function dv.

Note:

The function ‘u’ is differentiated up to constant.

Example  1:

\[\color{red}{Evaluate:\ \int x^3\ sin\ x\ dx}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[ ILATE \]
\[u= x^3\ \hspace{2cm}\ dv = sin\ x\ dx\]
\[v = – cos x\]
\[u^! = 3x^2\ \hspace{2cm}\ v_1 = – sin\ x\]
\[u^{!!} = 6x\ \hspace{2cm}\ v_2 = cos \ x\]
\[u^{!!!} = 6\ \hspace{2cm}\ v_3 = sin \ x\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 – u^{!!!}v_3+ c\]
\[\int x^3\ sin\ x\ dx = x^3(-cos x) – 3x^2 (- sin\ x) + 6x (cos\ x) – 6(sin x) + c\]
\[\boxed{\int x^3\ sin\ x\ dx = – x^3 cos x + 3x^2 sin\ x + 6x cos\ x – 6sin\ x+c}\]

Example  2:

\[\color{red}{Evaluate:\ \int x^3\ cos\ x\ dx}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[ ILATE \]
\[u= x^3\ \hspace{2cm}\ dv = cos\ x\ dx\]
\[ v = sin\ x\]
\[u^! = 3x^2\ \hspace{2cm}\ v_1 = – cos\ x\]
\[u^{!!} = 6x\ \hspace{2cm}\ v_2 = – sin \ x\]
\[u^{!!!} = 6\ \hspace{2cm}\ v_3 = cos \ x\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 – u^{!!!}v_3 + c\]
\[\int x^3\ cos\ x\ dx = x^3(sin x) – 3x^2 (- cos\ x) + 6x (- sin\ x) – 6( cos\ x)+c\]
\[\boxed{\int x^3\ cos\ x\ dx = x^3 sin x + 3x^2 cos\ x – 6x\ sin\ x – 6 cos\ x+c}\]

Example  3:

\[\color{red}{Evaluate:\ \int x^2\ sin\ 2x\ dx}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[ ILATE \]
\[u= x^2\ \hspace{2cm}\ dv = sin\ 2x\ dx\]
\[u^! = 2x\ \hspace{2cm}\ v = – \frac{cos\ 2x}{2}\]
\[u^{!!} = 2\ \hspace{2cm}\ v_1 = – \frac{sin\ 2x}{4}\]
\[\hspace{3cm}\ v_2 = \frac{cos\ 2x}{8}\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 \]
\[\int x^2\ sin\ 2x\ dx = x^2(-\frac{cos\ 2x}{2}) – 2x (-\frac{sin\ 2x}{4}) + 2(\frac{cos\ 2x}{8}) + c\]
\[ = – x^2\frac{cos\ 2x}{2} + x \frac{sin\ 2x}{2} + \frac{cos\ 2x}{4} + c\]
\[\boxed{\int x^2\ sin\ 2x\ dx = – x^2\frac{cos\ 2x}{2} + x \frac{sin\ 2x}{2} + \frac{cos\ 2x}{4}+c}\]

Example  4:

\[\color{red}{Evaluate:\ \int x^2\ cos\ 2x\ dx}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[ ILATE \]
\[u= x^2\ \hspace{2cm}\ dv = cos\ 2x\ dx\]
\[u^! = 2x\ \hspace{2cm}\ v = \frac{sin\ 2x}{2}\]
\[u^{!!} = 2\ \hspace{2cm}\ v_1 = – \frac{cos\ 2x}{4}\]
\[\hspace{3cm}\ v_2 = -\frac{sin\ 2x}{8}\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 \]
\[\int x^2\ cos\ 2x\ dx = x^2(\frac{sin\ 2x}{2}) – 2x (-\frac{cos\ 2x}{4}) + 2(- \frac{sin\ 2x}{8}) + c\]
\[ = x^2\frac{sin\ 2x}{2} + x \frac{cos\ 2x}{2} – \frac{sin\ 2x}{8} + c\]
\[\boxed{\int x^2\ cos\ 2x\ dx = x^2\frac{sin\ 2x}{2} + x \frac{cos\ 2x}{2} – \frac{sin\ 2x}{8}+c}\]

Example  5:

\[\color{red}{Evaluate:\ \int x^2\ e^{5x}\ dx}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[ ILATE \]
\[u= x^2\ \hspace{2cm}\ dv = e^{5x}\ dx\]
\[u^! = 2x\ \hspace{2cm}\ v = \frac{e^{5x}}{5}\]
\[u^{!!} = 2\ \hspace{2cm}\ v_1 = \frac{e^{5x}}{25}\]
\[\hspace{3cm}\ v_2 = \frac{e^{5x}}{125}\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 \]
\[\int x^2\ e^{5x}\ dx = x^2\frac{e^{5x}}{5}\ -\ 2\ x \frac{e^{5x}}{25}\ +\ 2\frac{e^{5x}}{125}\ +\ c\]

Example  6

\[\color{red}{Evaluate:\ \int x^2\ e^{-9x}\ dx}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[ ILATE \]
\[u= x^2\ \hspace{2cm}\ dv = e^{-9x}\ dx\]
\[u^! = 2x\ \hspace{2cm}\ v = \frac{e^{-9x}}{-9}\]
\[u^{!!} = 2\ \hspace{2cm}\ v_1 = \frac{e^{-9x}}{81}\]
\[\hspace{3cm}\ v_2\ =\ -\frac{e^{-9x}}{729}\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 \]
\[\int x^2\ e^{-9x}\ dx = x^2\frac{e^{-9x}}{-9}\ -\ 2\ x \frac{e^{-9x}}{81}\ +\ 2\frac{e^{-9x}}{-729}\ +\ c\]
\[ = -\ x^2\frac{e^{-9x}}{9}\ -\ 2x \frac{e^{-2x}}{81}\ -\ 2\frac{e^{-9x}}{729} + c\]
\[\boxed{\int x^2\ e^{-2x}\ dx =-\ x^2\frac{e^{-9x}}{9}\ -\ 2x \frac{e^{-2x}}{81}\ -\ 2\frac{e^{-9x}}{729}+c}\]

Example  7:

\[\color{red}{Evaluate:\ \int x^3\ e^{2x}\ dx}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[ ILATE \]
\[u= x^3\ \hspace{2cm}\ dv = e^{2x}\ dx\]
\[u^! = 3x^2\ \hspace{2cm}\ v = \frac{e^{2x}}{2}\]
\[u^{!!} = 6x\ \hspace{2cm}\ v_1 = \frac{e^{2x}}{4}\]
\[u^{!!!} = 6\ \hspace{2cm}\ v_2 = \frac{e^{2x}}{8}\]
\[\hspace{3cm}\ v_3 = \frac{e^{2x}}{16}\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 – u^{!!!}v_3\]
\[\int x^3\ e^{2x}\ dx = x^3\frac{e^{2x}}{2} – 3\ x^2 \frac{e^{2x}}{4} + 6x\frac{e^{2x}}{8} – 6\frac{e^{2x}}{16} + c\]
\[ = x^3\frac{e^{2x}}{2} – 3\ x^2 \frac{e^{2x}}{4} + 3x\frac{e^{2x}}{4} – 3\frac{e^{2x}}{8} + c\]
\[\boxed{\int x^3\ e^{2x}\ dx = x^3\frac{e^{2x}}{2} – 3\ x^2 \frac{e^{2x}}{4} + 3x\frac{e^{2x}}{4} – 3\frac{e^{2x}}{8}+c}\]

Example  8:

\[\color{red}{Evaluate:\ \int x^3\ e^{-3x}\ dx}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[ ILATE \]
\[u= x^3\ \hspace{2cm}\ dv = e^{-3x}\ dx\]
\[u^! = 3x^2\ \hspace{2cm}\ v =\ -\ \frac{e^{-3x}}{3}\]
\[u^{!!} = 6x\ \hspace{2cm}\ v_1 = \frac{e^{-3x}}{9}\]
\[u^{!!!} = 6\ \hspace{2cm}\ v_2 =\ -\ \frac{e^{-3x}}{27}\]
\[\hspace{3cm}\ v_3 = \frac{e^{-3x}}{81}\]
\[\int u\ dv = uv – u^!v_1 + u^{!!}v_2 – u^{!!!}v_3\]
\[\int x^3\ e^{-3x}\ dx = x^3(-\ \frac{e^{-3x}}{3})\ -\ 3\ x^2 (\frac{e^{-3x}}{9})\ +\ 6x(\frac{e^{-3x}}{27})\ -\ 6(\frac{e^{-3x}}{81}) + c\]
\[ =\ -\ x^3\frac{e^{-3x}}{3}\ -\ x^2 \frac{e^{-3x}}{3}\ -\ 2x\frac{e^{-3x}}{9}\ -\ 2\frac{e^{-3x}}{27} + c\]
\[\boxed{\int x^3\ e^{-3x}\ dx = -\ x^3\frac{e^{-3x}}{3}\ -\ x^2 \frac{e^{-3x}}{3}\ -\ 2x\frac{e^{-3x}}{9}\ -\ 2\frac{e^{-3x}}{27}+c}\]

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