منتديات الرياضيات العربية - عرض مشاركة واحدة - مسابقة صيف 2009

11-08-2009, 09:21 PM

أول أربع تكاملات ويمكن أن تحل بالتجزيء أو بالاعداد المركبة

122 )

$\\\int e^x\,sin\,x\,dx \\ \\=\int e^x\,\frac{e^{ix}-e^{-ix}}{2i} \\ \\=\frac{1}{2i}\int\left ( e^{x(1+i)}-e^{x(1-i) \right ) }\,dx \\ \\=\frac{1}{2i}\left (\frac{ e^{x(1+i)}}{1+i}-\frac{e^{x(1-i)}}{1-i} \right )+c \\ \\=\frac{e^x}{2}\left(\frac{e^{ix}-e^{-ix}-ie^{ix}-ie^{-ix}}{2i} \right )+c \\ \\= \frac{e^x}{2}\left(\frac{e^{ix}-e^{-ix}}{2i}-\frac{e^{ix}+e^{-ix}}{2} \right )+c \\ \\= \frac{e^x(sin\,x-cos\,x)}{2}+c$

123 )

$\\\int e^{ax}\,sin\,nx\,dx \\ \\=\int e^{ax}\,\frac{e^{inx}-e^{-inx}}{2i} \\ \\=\frac{1}{2i}\int\left ( e^{x(a+in)}-e^{x(a-in) \right ) }\,dx \\ \\=\frac{1}{2i}\left (\frac{ e^{x(a+in)}}{a+in}-\frac{e^{x(a-in)}}{a-in} \right )+c \\ \\=\frac{e^{ax}}{a^2+n^2}\left(\frac{ae^{inx}-ae^{-inx}-ine^{ix}-ine^{-ix}}{2i} \right )+c \\ \\= \frac{e^{ax}}{a^2+n^2}\left(a\left( \frac{e^{inx}-e^{-inx}}{2i}\right )-n\left( \frac{e^{inx}+e^{-inx}}{2}\right ) \right )+c \\ \\= \frac{e^{ax}(asin\,x-ncos\,x)}{a^2+n^2}+c$

124 )

$\\ \int e^{x}\,cos\,x\,dx \\ \\u = e^x \; \; \; \; du=e^xdx \\dv= cos\,x\,dx \; \; \; \; v= sin\,x \\ \\ \int e^{x}\,cos\,x\,dx= e^x\,sin\,x-\int e^x\,sin\,x\,dx \\ \\ u = e^x \; \; \; \; du =e^x dx \\dv=sin\,x \; \; \; \; v=-cos\,x \\ \\ \int e^{x}\,cos\,x\,dx= e^x\,sin\,x+e^x\,cos\,x-\int e^{x}\,cos\,x\,dx \\ \\\int e^{x}\,cos\,x\,dx=\frac{e^x}{2}\left(sin\,x+cos\,x \right )+c$

125 )

$\\ \int e^{ax}\,cos\,nx\,dx \\ \\u = e^{ax} \; \; \; \; du=ae^{ax}dx \\ \\dv= cos\,nx\,dx \; \; \; \; v= \frac{1}{n}sin\,nx \\ \\ \int e^{ax}\,cos\,nx\,dx=\frac{1}{n} e^{ax}\,sin\,nx-\frac{a}{n}\int e^{ax}\,sin\,nx\,dx \\ \\ u = e^{ax} \; \; \; \; du =ae^{ax} dx \\ \\dv=sin\,nx \; \; \; \; v=-\frac{1}{n}cos\,nx \\ \\ \int e^{x}\,cos\,x\,dx= \frac{1}{n} e^{ax}\,sin\,nx+\frac{a}{n^2}e^{ax}\,cos\,nx-\frac{a^2}{n^2}\int e^{x}\,cos\,x\,dx \\ \\ \int e^{ax}\,cos\,nx\,dx = \frac{e^{ax}\left( \frac{1}{n} \,sin\,nx+\frac{a}{n^2}\,cos\,nx\right ) }{1+\frac{a^2}{n^2}}+c \\ \\ \int e^{ax}\,cos\,nx\,dx = \frac{e^{ax}\left(n \,sin\,nx+a\,cos\,nx\right ) }{n^2+a^2} +c$