integration

مهيب الأرنؤوطى · #1 02-11-2011, 05:44 PM

$\underline{integration\, is\, the\: last\: chapter\: in\,\, calculus}$

$let\, f\, be\, a\, function\, and\, {g}'\left ( x \right )=f\left ( x \right ),\, then\: g\left ( x \right )\, is\, called\, the\, inverse\, derivative$

$i.e\, \, g\left ( x \right )=\int f\left ( x \right )\cdot dx\, \, if\, and\, only\, if\, {g}'\left ( x \right )=f\left ( x \right )$
$of\, the\, funcion\, \left ( f \right )\, \,$

$important\, rules:$


	$\int x^n\cdot dx=\frac{x^n^+^1}{n+1}+c,\, n\neq -1\, and\, \left ( c \right )\, is\, constant$

$examples:$

$( a \right )\, \int 4x^3\cdot dx=4\times \frac{x^4}{4}+c=x^4+c$

$\left ( b \right )\int \frac{-4}{3\sqrt{x}}\cdot dx=\frac{-4}{3}\int \frac{1}{x^\frac{1}{2}}\cdot dx=\frac{-4}{3}\int x^{-\frac{1}{2}}\cdot dx=\frac{-4}{3}\times\frac{x^\frac{1}{2}}{\frac{1}{2}}+c$

مهيب الأرنؤوطى · #2 02-11-2011, 07:04 PM


	$theorem\, \, 2:$


	$if\, \, a\, \, and\, \, b\, \, are\, \, constants,\, \, and\, \, n\neq -1,\, \, then:$ $\int \left ( ax+b \right )^n\cdot dx=\frac{\left ( ax+b \right )^n^+^1}{a\left ( n+1 \right )}+c$

$examples:$

$\left ( a \right )\, \int \left ( 2x+5 \right )^3dx=\frac{\left ( 2x+5 \right )^4}{2\times 4}+c$

$\left ( b \right )\, \int\frac{8dx}{\left ( 3-5x \right )^5}$

$=\int 8\left ( 3-5x \right )^-^5dx=\frac{8\left ( 3-2x \right )^-^4}{-2\left ( -4 \right )}+c$

مهيب الأرنؤوطى · #3 02-11-2011, 11:59 PM

$\underline{some\, \, examples:}$

$\left ( a \right )\, \, \int \sqrt{2x+7}dx$

$=\int \left ( 2x+7 \right )^\frac{1}{2}dx\, \, =\frac{\left ( 2x+7 \right )^\frac{3}{2}}{2\times \frac{3}{2}}+c\, \, =\frac{\sqrt{\left ( 2x+7 \right )^3}}{3}+c$

***************

$\left ( b\, \right )\, \,\int \left ( 2x+3 \right ) \sqrt{x-1}dx$

$=\int \left \left \{ 2\left ( x-1 \right )+5 \right \}\left ( x-1 \right )^\frac{1}{2}dx$

$=2\int \left ( x-1 \right )^\frac{3}{2}dx+5\int \left ( x-1 \right )^\frac{1}{2}dx$

$=2\times \frac{\left ( x-1 \right )^\frac{5}{2}}{\frac{5}{2}}+5\times \frac{\left ( x-1 \right )^\frac{5}{2}}{\frac{5}{2}}+c$

***************

$\left ( c \right )\, \, \int \frac{3x+7}{\left ( x+2 \right )^4}dx$

$=\int \frac{3\left ( x+2 \right )+1}{\left ( x+2 \right )^4}dx$

$=3\int \left (x+2 \right )^-^3dx+\int \left ( x+2 \right )^-^4dx$

$=\frac{\left ( x+2 \right )^-^2}{-2\times 3}+\frac{\left ( x+2 \right )^-^3}{-3}+c$

***************

$\left ( d \right )\, \int \frac{1}{x}\sqrt{x^3+x^2}dx$

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

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

**الاستاذ على الدين يحيى** · #4 02-12-2011, 08:00 PM

اقتباس:

المشاركة الأصلية كتبت بواسطة مهيب الأرنؤوطى

$\left ( d \right )\, \int \frac{1}{x}\sqrt{x^3+x^2}dx$

$=\int \frac{1}{x^2}\sqrt{x^3+x^2}dx$

بعد إذن حضرتك
أعتقد أن حضرتك فى هذه الخطوة تقصد أن
1 / x تصبح 1 / x^2 داخل الجذر التربيعى وليس خارجه كما هى موضحة .

نشكرك على عملك الرائع وبارك الله فيك

مهيب الأرنؤوطى · #5 02-12-2011, 09:16 PM

أجل أستاذي الفاضل، فهذا ما أقصده بالضبط ولكنها زلة قلم ليس إلا، وعلي العموم فسوف أقوم بتعديلها حالاً....

أشكرك علي التنبيه

مهيب الأرنؤوطى · #6 02-13-2011, 12:18 AM

$remember\, \, that:$

$\left ( 1 \right )\frac{\mathrm{d} y}{\mathrm{d} x}\, \, is\, \, the\, \, slope\, \, of\, \, the\, \, tengant\, \, at\, \, any\, point\, \, on\, \, the\, \, curve$

$\left ( 2 \right )if:\, \,x _{1}=a\, \, and\, \,x _{2}=b\, \, are\, \, critical\, \, points:$

$\left ( a \right )if\, \, {f}''\left ( a)> 0,\, \, then\, \, there\, \, is\, \, a\, \, local\, \, min.$

$\left ( b \right )if\, \, {f}''\left ( a)< 0,\, \, then\, \, there\, \, is\, \, a\, \, local\, \, max.$

$\left ( 3 \right )the\, \, slope\, \, of\, \, tengant\, \, whose\, \, its\, \, eqation\, \, is:$

$ax+by+c=0,\, \, is:\frac{-a}{b}$

$\left ( 4 \right )L_{1}\parallel L_{2}\, \, iff\, \, m_{1}=m_{2}:\, m\, \, is\, \, the\, \, slope$

$\left ( 4 \right )L_{1}\perp L_{2}\, \, iff\, \, m_{1}\cdot m_{2}=-1\, \,: m\, \, is\, \, the\, \, slope$

$\left ( 6 \right )tan\theta =m:\theta \, \, is\, \, the\, \, angle\, \, between\, \, the\, \, tengant\, \, and\, \, the\, \, positve\, \, dirction\, \, of\, \, X-axis$

$\left ( 7 \right )\, \, \frac{\mathrm{d^2} y}{\mathrm{d} x^2}\, \, is\, \, the\, \, rate\, \, of\, \, change\, of\, \, the\, \, slope\, of\, \, tengant$

مهيب الأرنؤوطى · #7 02-13-2011, 12:35 AM

$\underline{important\, \, rules:}$

$\left ( a \right )\int sin\theta \cdot d\theta =-cos\theta +c$

$\left ( b \right )\int cos\theta \cdot d\theta= sin\theta +c$

$\left ( c \right )\int sec^\theta \theta \cdot d\theta= tan\theta +c$

$\left (d \right )\int cosec^2\theta \cdot d\theta=-cot\theta +c$

*****************

$\left ( e \right )\int sin\left ( ax \right )\cdot dx=\frac{-1}{a}cos\left ( ax \right )+c$

$\left ( f \right )\int cos\left ( ax \right )\cdot dx=\frac{1}{a}sin\left ( ax \right )+c$

$\left ( g \right )\int sec^2\left ( ax \right )\cdot dx=\frac{1}{a}tan\left ( ax \right )+c$

$\left ( h\right )\int cosec^2\left ( ax \right )\cdot dx=\frac{-1}{a}cot\left ( ax \right )+c$

$such\, \, a\,\, and\, \, c\, \, are\, \, constants$

مهيب الأرنؤوطى · #8 02-13-2011, 01:20 AM

$important\, \, rule:$


	$if\, \, \frac{\mathrm{d} y}{\mathrm{d} x}=f\left ( x \right )\cdot r\left ( y \right )\, \, then\, \, \int \frac{1}{r\left ( y \right )}dy=\int f\left ( x \right )\cdot dx$

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

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

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

${\color{red} solution}$

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

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

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

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

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

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

**************

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

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

${\color{red} solution}$

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

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

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

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

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

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

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

مهيب الأرنؤوطى · #10 02-13-2011, 09:40 PM

$\underline{some\, \, problems\, \, about\, \, integraion:}$

${\color{red} evaluate\, \, the\, \, following\, \, integrations:}$

$\left ( 1 \right )\, \int \sqrt{\left ( x^2-\frac{1}{x^2} \right )^2+4}\cdot dx$

$\left ( 2\, \right )\, \, \int \left ( 2x-1 \right )\left ( x-1 \right )^7\cdot dx$

$\left ( 3 \right )\, \, if\, y=x\sqrt{x+1},\, \, prove\, \, that\, \, \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{3x+2}{2\sqrt{x+1}},\, \, and\, \, hence,\, find:\, \int \frac{3x+4}{2\sqrt{x+1}}$

$\left ( 4 \right )\, prove\, \, that\, \int \frac{2x}{\left ( x^2+1 \right )}\cdot dx=\frac{x^2}{x^2+1}+c$

$\left ( 5 \right )\, find\, \, a\left ( h \right )\, \, if\, \, a\left ( h \right )=\int \sqrt{x^2-4x+4}\cdot dx\, :\, a\left ( 2 \right )=3\, \, and\, \, \left ( a \right )\, \, is\, \, continous\, \, at\, \, R$

$\left ( 6 \right )\, \int \left ( 4x^2-20x+25 \right )^\frac{7}{2}\cdot dx$

$\left ( 7 \right )\, \int \left ( \sqrt{x^3}+\frac{1}{\sqrt{x^3}} \right )^2\cdot sin^2a\cdot dx$

$\left ( 8 \right )\, \int \frac{3x^3-24x^2+48x+5}{x^2-8x+16}\cdot dx$

$\left ( 9 \right )\, \int \frac{cos^22x}{1+sin2x}\cdot dx$

$\left ( 10 \right )\, \int \frac{x^4+x^2+1}{x^2+x+1}\cdot dx$

$\left ( 11 \right )\, \int \frac{\left ( 1+x \right )^3-\left ( 1-x \right )^3}{\left ( 1-x^2 \right )^3}\cdot dx$

$\left ( 12 \right )\, if\, \, f\left ( x \right )\, is\, \, satisfied\, \, by\, \, the\, \, relation:\, f\left ( a+h \right )-f\left ( a \right )=c\cdot a\cdot h+2h^2\, \, for\, \, all\, \, a\, \, and\, \, h,\, s\cdot t\, \left ( c \right )\, is\, \, constant,\, \, and\, \, f\left ( 1 \right )=7\, \, and\, \, f\left ( 2 \right )=13,\, \, find\, \, f\left ( x \right )\cdot$

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