Упр.356 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1. 356. Найти первообразные для функции:
   1) $$2x^5-3x^2$$;
   2) $$5x^4+2x^3$$;
   3) $$3x^3+2x-1$$;
   4) $$6x^2-4x+3$$;
   5) $$\frac{2}{x}+\frac{3}{x^2}$$;
   6) $$4\sqrt[3]{x}-6\sqrt{x}$$;
   7) $$\sqrt{x}+2\sqrt[3]{x}$$;
   8) $$\frac{2}{x^3}-\frac{3}{x}$$.

1. $$f(x)=2x^5-3x^2$$

   $$F(x)=2\cdot \frac{x^6}{6}-3\cdot \frac{x^3}{3}+C=\frac{x^6}{3}-x^3+C$$

2. $$f(x)=5x^4+2x^3$$

   $$F(x)=5\cdot \frac{x^5}{5}+2\cdot \frac{x^4}{4}+C=x^5+\frac{x^4}{2}+C$$

3. $$f(x)=3x^3+2x-1$$

   $$F(x)=3\cdot \frac{x^4}{4}+2\cdot \frac{x^2}{2}-x+C=\frac{3x^4}{4}+x^2-x+C$$

4. $$f(x)=6x^2-4x+3$$

   $$F(x)=6\cdot \frac{x^3}{3}-4\cdot \frac{x^2}{2}+3x+C=2x^3-2x^2+3x+C$$

5. $$f(x)=\frac{2}{x}+\frac{3}{x^2}=2x^{-1}+3x^{-2}$$

   $$F(x)=2\ln|x|+3\cdot \frac{x^{-1}}{-1}+C=2\ln|x|-\frac{3}{x}+C$$

6. $$f(x)=4\sqrt[3]{x}-6\sqrt{x}=4x^{1/3}-6x^{1/2}$$

   $$F(x)=4\cdot \frac{x^{4/3}}{4/3}-6\cdot \frac{x^{3/2}}{3/2}+C=3x^{4/3}-4x^{3/2}+C$$

7. $$f(x)=\sqrt{x}+2\sqrt[3]{x}=x^{1/2}+2x^{1/3}$$

   $$F(x)=\frac{x^{3/2}}{3/2}+2\cdot \frac{x^{4/3}}{4/3}+C=\frac{2}{3}x^{3/2}+\frac{3}{2}x^{4/3}+C$$

8. $$f(x)=\frac{2}{x^3}-\frac{3}{x}=2x^{-3}-3x^{-1}$$

   $$F(x)=2\cdot \frac{x^{-2}}{-2}-3\ln|x|+C=-\frac{1}{x^2}-3\ln|x|+C$$

Ответ

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