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:

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

$\color{purple} {Example\ 9.}\ \color{red}{Evaluate\ :}\ \int x^3\ e^{-x}\ dx\ \hspace{15cm}$
$\color {blue}{Soln:}\ \hspace{20cm}$
$ILATE$
$u= x^3\ \hspace{2cm}\ dv = e^{-x}\ dx$
$u^! = 3x^2\ \hspace{2cm}\ v =\ -\ e^{-x}$
$u^{!!} = 6x\ \hspace{2cm}\ v_1\ =\ e^{-x}$
$u^{!!!} = 6\ \hspace{2cm}\ v_2 =\ -\ e^{-x}$
$\hspace{3cm}\ v_3\ =\ e^{-x}$
$\int u\ dv = uv – u^!v_1 + u^{!!}v_2 – u^{!!!}v_3$
$\int x^3\ e^{-x}\ dx\ =\ x^3(-\ e^{-x})\ -\ 3\ x^2 (e^{-x})\ +\ 6x(-\ e^{-x})\ -\ 6(e^{-x}) + c$
$=\ -\ x^3\ e^{-x}\ -\ 3\ x^2\ e^{-x}\ -\ 6x\ e^{-x}\ -\ 6e^{-x}\ +\ c$

Example  10:

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