Вычислите определённый интеграл:

Решите задачу:

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