integration

مهيب الأرنؤوطى · #9 02-13-2011, 03:23 PM

$\underline{examples\, \, on\, \, the\, \, previous\, \, rule:}$

$\left ( 1 \right )let\, \, \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{1}{\sqrt{3-2x}}\, and\, \, y=3\, \, when\, \, x=1,\, \, then\, \, find\, \, the\, \, relation\, \, between\, \, x\, \, and\, \, y$

${\color{red} solution}$

$\because \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{1}{\sqrt{3-2x}}$

$\therefore \frac{\mathrm{d} y}{\mathrm{d} x}=\left ( 3-2x \right )^\frac{-1}{2}$

$\therefore y=\int \left ( 3-2x \right )^\frac{-1}{2}\cdot dx=-\sqrt{3-2x}+c$

$But\, \, y=3\, \, when\, \, x=1\Rightarrow 3=-\sqrt{3-2}+c\Rightarrow c=4$

$\therefore the\, \, relation\, \, between\, \, x\, \, and\, \, y\, \, is:$

$y=-\sqrt{3-2x}+4$

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$\left ( 2 \right )\, if\, \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{1-sin^2y}{1-sin^22x},\, \, find\, \, the\, \, relation\, \, between:$

$x\, \, and\, \, y\, \, knowing\, \, that\, \, y=\frac{3\pi }{4}\, when\, x=\frac{\pi }{8}$

${\color{red} solution}$

$\because \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{1-sin^2y}{1-sin^22x}\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{cos^2y}{cos^22x}$

$\therefore \frac{1}{cos^2y}\cdot dy=\frac{1}{cos^22x}\cdot dx\Rightarrow sec^2y\cdot dy=sec^22x\cdot dx$

$\therefore\, \int sec^2y\cdot dy=\int sec^22x\cdot dx$

$\therefore tany=\frac{1}{2}tan2x+c$

$\because y=\frac{3\pi }{4}\, \, when\, \, x=\frac{\pi }{8}\Rightarrow tan\frac{3\pi }{4}=\frac{1}{2}tan\frac{\pi }{4}+c$

$\therefore -1=\frac{1}{2}+c$

$c=\frac{-3}{2}\Rightarrow tany=\frac{1}{2}tan2x-\frac{3}{2}$

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