الموقع التخصصى فى الرياضيات للثانوية العامة - عرض مشاركة واحدة

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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