الموقع التخصصى فى الرياضيات للثانوية العامة - عرض مشاركة واحدة

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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