What is \(\displaystyle\int e^{x\ln 10}\,e^{x}\,dx\) equal to?
\(\dfrac{(10e)^{x}}{1+\ln 10}+c\)
Since \(e^{x\ln 10}=10^{x}\), the integral becomes \(\int 10^{x}e^{x}\,dx=\int (10e)^{x}\,dx\). Using \(\int a^{x}dx=\dfrac{a^{x}}{\ln a}+c\) with \(a=10e\) and \(\ln(10e)=1+\ln 10\), the result is \(\dfrac{(10e)^{x}}{1+\ln 10}+c\).
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