Question

100 баллов 2 примера ​

Answer 1

$1)\ \ f(x)=\dfrac{1}{cos^2(2x-\frac{\pi}{4})}\\\\\\F(x)=\int \dfrac{dx}{cos^2(2x-\frac{\pi}{4})}=\dfrac{1}{2}\, tg(2x-\frac{\pi}{4})+C\\\\\\F(\dfrac{\pi}{4})=1\ \ \to \ \ \ F(\dfrac{\pi}{4})=\dfrac{1}{2}\cdot tg\dfrac{\pi}{4}+C=1\ \ ,\ \ \ \dfrac{1}{2}+C=1\ \ ,\ \ C=\dfrac{1}{2}\\\\\\F(x)\Big|_{(\frac{\pi}{4};1)}=\dfrac{1}{2}\, tg(2x-\frac{\pi}{4})+\dfrac{1}{2}$

$2)\ \ y=x^2+4x+4\ \ \to \ \ \ y=(x+2)^2\ \ ,\ \ x=0\ ,\ y=0\ ,\ x=-3\\\\S=\int\limits_{-3}^0\, (x+2)^2\, dx=\dfrac{(x+2)^3}{3}\, \Big|_{-3}^0=\dfrac{1}{3}\cdot (2^3-(-1)^3)=\dfrac{1}{3}\cdot 9=3$