Упр.364 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1) f(x) = 2x + 3, М(1; 2);
2) f(x) = 4x-1, М(-1; 3);
3) f(x)= корень x +2, М(2; -3);
4) f(x)= 1/корень x+3, М(-2; -1);
5) f(x) = sin2х, М(пи/2; 5);
6) f(x) = cos3x, М(0; 0);
7) f(х) = 1/x+3, М(-2; 4);
8) f(x) = 1/(x+1)2, М(-2; 2).
* + 3 (х+1)2

1. $$f(x)=2x+3,\quad M(1;2)$$

   Первообразная:

   $$F(x)=\int (2x+3)\,dx=x^2+3x+C$$

   Подставим точку $$M(1;2)$$:

   $$2=1^2+3\cdot 1+C$$

   $$C=-2$$

   $$F(x)=x^2+3x-2$$

2. $$f(x)=4x-1,\quad M(-1;3)$$

   $$F(x)=\int (4x-1)\,dx=2x^2-x+C$$

   $$3=2\cdot(-1)^2-(-1)+C$$

   $$C=0$$

   $$F(x)=2x^2-x$$

3. $$f(x)=\sqrt{x+2},\quad M(2;-3)$$

   $$F(x)=\int (x+2)^{1/2}\,dx=\frac{2}{3}(x+2)^{3/2}+C$$

   $$-3=\frac{2}{3}(2+2)^{3/2}+C$$

   $$-3=\frac{2}{3}\cdot 8+C$$

   $$C=-\frac{25}{3}$$

   $$F(x)=\frac{2}{3}(x+2)^{3/2}-\frac{25}{3}$$

4. $$f(x)=\frac{1}{\sqrt{x+3}},\quad M(-2;-1)$$

   $$F(x)=\int (x+3)^{-1/2}\,dx=2\sqrt{x+3}+C$$

   $$-1=2\sqrt{-2+3}+C$$

   $$-1=2+C$$

   $$C=-3$$

   $$F(x)=2\sqrt{x+3}-3$$

5. $$f(x)=\sin 2x,\quad M\left(\frac{\pi}{2};5\right)$$

   $$F(x)=\int \sin 2x\,dx=-\frac{1}{2}\cos 2x+C$$

   $$5=-\frac{1}{2}\cos \pi +C$$

   $$5=\frac{1}{2}+C$$

   $$C=\frac{9}{2}$$

   $$F(x)=-\frac{1}{2}\cos 2x+\frac{9}{2}$$

6. $$f(x)=\cos 3x,\quad M(0;0)$$

   $$F(x)=\int \cos 3x\,dx=\frac{1}{3}\sin 3x+C$$

   $$0=\frac{1}{3}\sin 0+C$$

   $$C=0$$

   $$F(x)=\frac{1}{3}\sin 3x$$

7. $$f(x)=\frac{1}{x+3},\quad M(-2;4)$$

   $$F(x)=\int \frac{1}{x+3}\,dx=\ln|x+3|+C$$

   $$4=\ln|-2+3|+C$$

   $$4=C$$

   $$F(x)=\ln|x+3|+4$$

8. $$f(x)=\frac{1}{(x+1)^2},\quad M(-2;2)$$

   $$F(x)=\int (x+1)^{-2}\,dx=-\frac{1}{x+1}+C$$

   $$2=-\frac{1}{-2+1}+C$$

   $$2=1+C$$

   $$C=1$$

   $$F(x)=-\frac{1}{x+1}+1$$

Ответ

1. $$F(x)=x^2+3x-2$$
2. $$F(x)=2x^2-x$$
3. $$F(x)=\frac{2}{3}(x+2)^{3/2}-\frac{25}{3}$$
4. $$F(x)=2\sqrt{x+3}-3$$
5. $$F(x)=-\frac{1}{2}\cos 2x+\frac{9}{2}$$
6. $$F(x)=\frac{1}{3}\sin 3x$$
7. $$F(x)=\ln|x+3|+4$$
8. $$F(x)=-\frac{1}{x+1}+1$$