Упр.24.16 ГДЗ Мордкович Семенов 11 класс (Алгебра)

а) f(x)={x^3+1, -1×1; x^2-4x+5, 1

1. а)

   $$S=\int_{-1}^{1}(x^3+1)\,dx+\int_{1}^{4}(x^2-4x+5)\,dx.$$

   $$S=\left(\frac{x^4}{4}+x\right)\Big|_{-1}^{1}+\left(\frac{x^3}{3}-2x^2+5x\right)\Big|_{1}^{4}.$$

   $$S=\left(\frac14+1-\frac14+1\right)+\left(\frac{64}{3}-32+20-\frac13+2-5\right)=2+6=8.$$

2. б)

   $$S=\int_{0}^{1}2x\,dx+\int_{1}^{2}\frac{2}{x^2}\,dx.$$

   $$S=x^2\Big|_{0}^{1}-\frac{2}{x}\Big|_{1}^{2}=1+\left(-1+2\right)=2.$$

3. в)

   $$S=\int_{-1}^{1}0{,}5^x\,dx+\int_{1}^{2}\frac{1}{2x}\,dx.$$

   $$S=\frac{0{,}5^x}{\ln 0{,}5}\Big|_{-1}^{1}+\frac12\ln|x|\Big|_{1}^{2}.$$

   $$S=\frac{0{,}5-2}{\ln 0{,}5}+\frac12\ln 2=\frac{3}{2\ln 2}+\frac12\ln 2.$$

4. г)

   $$S=\int_{1}^{2}x^3\,dx+\int_{2}^{4}(x^2-8x+20)\,dx.$$

   $$S=\frac{x^4}{4}\Big|_{1}^{2}+\left(\frac{x^3}{3}-4x^2+20x\right)\Big|_{2}^{4}.$$

   $$S=\left(4-\frac14\right)+\left(\frac{64}{3}-64+80-\frac83+16-40\right)=\frac{15}{4}+\frac{53}{12}=\frac{14}{3}.$$

5. д)

   $$S=\int_{-3}^{0}\sqrt{x+4}\,dx+\int_{0}^{1}2(x-1)^2\,dx.$$

   $$S=\frac{2}{3}(x+4)^{3/2}\Big|_{-3}^{0}+\frac{2}{3}(x-1)^3\Big|_{0}^{1}.$$

   $$S=\frac{2}{3}(8-1)+\frac{2}{3}(0-(-1))=\frac{14}{3}+\frac{2}{3}=\frac{16}{3}.$$

6. е)

   $$S=\int_{0}^{1}2^x\,dx+\int_{1}^{e}\frac{2}{x}\,dx.$$

   $$S=\frac{2^x}{\ln 2}\Big|_{0}^{1}+2\ln x\Big|_{1}^{e}.$$

   $$S=\frac{2-1}{\ln 2}+2(1-0)=2+\frac{1}{\ln 2}.$$

Ответ

а) $$8$$; б) $$2$$; в) $$\frac12\ln 2+\frac{3}{2\ln 2}$$; г) $$\frac{14}{3}$$; д) $$\frac{16}{3}$$; е) $$2+\frac{1}{\ln 2}$$.