Упр.542 ГДЗ Колмогоров 10-11 класс (Алгебра)

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Используем формулу

$$\int a^x\,dx=\frac{a^x}{\ln a}+C,\qquad a>0,\ a\ne 1.$$

1. $$\int_0^1 0{,}5^x\,dx=\left.\frac{0{,}5^x}{\ln 0{,}5}\right|_0^1=\frac{0{,}5-1}{\ln 0{,}5}=\frac{-\frac12}{-\ln 2}=\frac{1}{2\ln 2}.$$

2. $$\int_0^1 e^{2x}\,dx=\left.\frac{e^{2x}}{2}\right|_0^1=\frac{e^2-1}{2}.$$

3. $$\int_{-2}^1 2^x\,dx=\left.\frac{2^x}{\ln 2}\right|_{-2}^1=\frac{2-\frac14}{\ln 2}=\frac{7}{4\ln 2}.$$

4. $$\int_{1/2}^2 3^x\,dx=\left.\frac{3^x}{\ln 3}\right|_{1/2}^2=\frac{9-\sqrt3}{\ln 3}.$$

Ответ: а) $$\frac{1}{2\ln 2}$$; б) $$\frac{e^2-1}{2}$$; в) $$\frac{7}{4\ln 2}$$; г) $$\frac{9-\sqrt3}{\ln 3}$$.