Question

Вычислите определённый интеграл:

Answer 1

$displaystyle 1) int^3_1 frac{dx}{x^2}=int^3_1x^{-2}dx= frac{x^{-1}}{-1}=- frac{1}{x} bigg|^3_1=- frac{1}{3}+1= frac{2}{3}$
$displaystyle 2) int^2_{-1} 2x^3dx= frac{2x^4}{4}= frac{x^4}{2} bigg|^2_{-1}= frac{16}{2}- frac{1}{2}= frac{15}{2}$
$3) int^{pi}_{pi/2}sinxdx=-cosxbigg|^{pi}_{pi/2}=-cos pi+cos frac{pi}{2}=1$
$displaystyle 4) int^2_1 (3x^2+x-4)dx=x^3+ frac{x^2}{2}-4x bigg|^2_1=8+2-8-1- frac{1}{2} +4 = frac{9}{2}$
$displaystyle 5) int^4_1 frac{dx}{sqrt{x}}=int^4_1x^{-1/2}dx=2sqrt{x} bigg|^4_1=2cdot2-2cdot1=2$