Question

Помогите пожалуйста срочно

Answer 1

$1)\; \; f(x)=2(3x-1)^2\ \ ,\ \ \ F(x)=\dfrac{2(3x-1)^3}{9}+C\\\\\\2)\ \ \int\limits^8_0\, (\sqrt{2x}+\sqrt[3]{x})\, dx=\Big(\dfrac{\sqrt2\, x^{3/2}}{3/2}+\dfrac{x^{4/3}}{4/3}\Big)\Big|_0^8=\dfrac{2\sqrt2}{3}\cdot 8\sqrt8+\dfrac{3}{4}\cdot 8\sqrt[3]8=\dfrac{100}{3}$

$b)\ \ \int\limits^2_{-1} \, (x^2+2x+1)\, dx=\int\limits^2_{-1}\, (x+1)^2\, dx=\dfrac{(x+1)^3}{3}\Big|_{-1}^2=\dfrac{1}{3}\cdot (27-0)=\dfrac{27}{3}=9\\\\\\3)\ \ y=-6x\ \ ,\ \ y=0\ \ ,\ \ x=4\\\\S=-\int\limits^4_0\, (-6x)\, dx=\dfrac{6x^2}{2}\Big|_0^4=3\cdot 16=48$

$4)\ \ \int \Big(\dfrac{3}{x^2}-\dfrac{2}{\sqrt{x}}+\dfrac{4\sqrt[3]{x^2}}{x}\Big)\, dx=-\dfrac{3}{x}-4\sqrt{x}+\dfrac{12x^{\frac{2}{3}}}{2}+C$