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Question Number 69327 by mhmd last updated on 22/Sep/19

$$\underset{0}{\stackrel{1}{\int }}\frac{2^{2x+1}-5^{2x-1}}{{10}^x}dx=$$

Answered by MJS last updated on 22/Sep/19

$$\int \frac{2^{2x+1}-5^{2x-1}}{{10}^x}dx=2\int {\left(\frac{2}{5}\right)}^{x}dx-\frac{1}{5}\int {\left(\frac{5}{2}\right)}^{x}dx=$$$$=\frac{2}{\ln \frac{2}{5}}{\left(\frac{2}{5}\right)}^x-\frac{1}{5\ln \frac{5}{2}}{\left(\frac{5}{2}\right)}^x=\frac{1}{\ln 2-\ln 5}(2{\left(\frac{2}{5}\right)}^x+\frac{1}{5}{\left(\frac{5}{2}\right)}^x)+C$$$$\underset{0}{\stackrel{1}{\int }}...dx=\frac{9}{10(\ln 5-\ln 2)}$$

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