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Question Number 145942 by Khalmohmmad last updated on 09/Jul/21

Answered by Olaf_Thorendsen last updated on 09/Jul/21

$$$$$$f(x+2)+10f\left(x\right)=7f(x+1)$$$$f(x+2)-7f(x+1)+10f\left(x\right)=0$$$$r^2-7r+10=0$$$$(r-2)(r-5)=0$$$$f\left(x\right)=\lambda .2^x+\mu 5^x$$$$f\left(0\right)=\lambda +\mu =2$$$$f\left(1\right)=2\lambda +5\mu =7$$$$\lambda =1,\mu =1$$$$f\left(x\right)=2^x+5^x$$$$f\left(x\right)=e^{x\ln 2}+e^{x\ln 5}$$$${\int }_0^{1}f\left(x\right)dx={[\frac{2^x}{\ln 2}+\frac{5^x}{\ln 5}]}_0^1=\frac{1}{\ln 2}+\frac{4}{\ln 5}$$$${\int }_0^{1}f\left(x\right)dx=\frac{\ln 5+4\ln 2}{\ln 2.\ln 5}=\frac{\ln 80}{\ln 2.\ln 5}$$$$\ln 2.\ln 5{\int }_0^{1}f\left(x\right)dx=\ln 80$$

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