https://www.codecogs.com/latex/eqneditor.php
: 4
\[Find \int {\color{Red} e^x sin x}\: dx\]
\[Find \int {\color{Red} e^x sin x}\: dx\]
https://www.codecogs.com/latex/eqneditor.php
: 4
\[Find \int {\color{Red} e^x sin x}\: dx\]
\[Find \int {\color{Red} e^x sin x}\: dx\]
; 05-28-2016 11:16 AM
$\int {\color{Red} e^x sin x}\: dx$
$ Let\\ e^x =dv , \mapsto \space v= e^x ,u=sinx \mapsto du= cosx$
$\int {\color{Red} e^x sin x}\: dx= sinx \space e^x - \int {\color{Red} e^x cosx}\: dx$
$ Let\\\ e^x =dv , \mapsto \space v= e^x ,u=cosx \mapsto du= -sinx$
$\int {\color{Red} e^x sin x}\: dx= e^x sinx (e^xcosx+ \int {\color{Red} e^x sinx \: dx}) $
$\int {\color{Red} e^x sin x}\: dx=\frac{1}{2}( e^x sinx e^xcosx)$
; 05-28-2016 10:31 PM
(( ))
$\int {\color{Red} e}^{x}\sin xdx$
$\int e^{x}.\frac{e^{ix}-e^{-ix}}{2i}dx=\frac{1}{2i}\int [e^{(1+i)x}-e^{(1-i)x}]dx$
$=\frac{1}{2i}[\frac{e^{(1+i)x}}{1+i}-\frac{e^{(1-i)x}}{1-i}]+C$
$=\frac{e^{x}}{2i}[\frac{e^{ix}-ie^{ix}-e^{-ix}-ie^{-ix}}{2}]+C$
$=\frac{e^{x}}{2}[\frac{e^{ix}-e^{-ix}}{2i}-\frac{e^{ix}+e^{-ix}}{2}]+C$
$= \frac{e^{x}}{2}(\sin x-\cos x)+C$
; 05-29-2016 05:33 PM
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$\int e^x sinx dx$
$=\int( \frac{1}{2}e^x cosx-\frac{1}{2}e^x cosx+\frac{1}{2}e^x sinx+\frac{1}{2}e^x sinx)dx$
$=\int( \frac{1}{2}e^x (cosx+sinx)+\frac{1}{2}e^x (sinx-cosx))dx$
$\int d(\frac{1}{2}e^x (sinx-cosx))=\frac{1}{2}e^x (sinx-cosx)+c$
" "
$\int_{0}^{2} x\sqrt{4-2x}dx $
$=1/2\int_{0}^{2}\sqrt{4-2x}((4-(4-2x))dx$
$=\int_{0}^{2}(2\sqrt{4-2x}-1/2(4-2x)^{3/2})dx$
$=-2/3(4-2x)^{3/2}+1/10(4-2x)^{5/2}\left. \right \}_{0}^{2}$
$=32/15$
; 05-31-2016 12:57 AM
" "
$\int_{1}^{2}x\sqrt{3x-2}dx $
$=1/3\int_{1}^{2}\sqrt{3x-2}(2+3x-2)dx$
$=2/3\int_{1}^{2}\sqrt{3x-2}dx+1/3\int_{1}^{2}(3x-2)^{3/2}dx$
$=4/27(3x-2)^{3/2}+2/45(3x-2)^{5/2}]_{1}^{2}$
$=326/135$
" "
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