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TuringTestBest ResponseYou've already chosen the best response.1
\[\frac d{dx}a^x=a^x\ln a\]so then what is the integral?
 2 years ago

virtusBest ResponseYou've already chosen the best response.1
actually i don't know hahaha
 2 years ago

MikeyBest ResponseYou've already chosen the best response.0
The indefinite integral of a^x is a^x/lna + C
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.1
\[\int a^x\ln adx=a^x\]so\[\int a^xdx=\frac{a^x}{\ln a}\]+C if it is indefinite, but yours is definite anyway...
 2 years ago

virtusBest ResponseYou've already chosen the best response.1
turning test can you please show me how this was derived. Thank you very much!
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.1
do you need me to prove that\[\frac d{dx}a^x=a^x\ln a\]? that would be the most thorough way to start
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.1
we can prove it with logarithmic differentiation\[y=a^x\]taking the natural of of both sides\[\ln y=\ln(a^x)\]\[\ln y=x\ln a\]now differentiate implicitly\[\frac{y'}y=\ln a\]\[y'=y\ln a=a^x\ln a\]so what does this tell us about the antiderivative?
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.1
\[y'=a^x\ln a\iff y=a^x\]so\[\int a^x\ln a=a^x\]now we can use this info and do your integral with a usubstitution
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.1
watch the sub carefully:\[\int a^xdx\]\[u=a^x\]\[du=a^x\ln a dx\to\frac{du}{\ln a}=a^xdx\]subbing in our expresison for a^xdx we get\[\frac1{\ln a}\int du=\frac u{\ln a}+C=\frac{a^x}{\ln a}+C\]
 2 years ago

TuringTestBest ResponseYou've already chosen the best response.1
any questions about that?
 2 years ago

virtusBest ResponseYou've already chosen the best response.1
THANK YOU A MILLION TIMES OVER turning test! greatly appreciated
 2 years ago
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