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anonymous
 4 years ago
integrate y=2^x from x=0 to x=3
anonymous
 4 years ago
integrate y=2^x from x=0 to x=3

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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0actually i don't know hahaha

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

TuringTest
 4 years ago
Best 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...

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0turning test can you please show me how this was derived. Thank you very much!

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

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.1we 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?

TuringTest
 4 years ago
Best 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

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.1watch 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\[\frac1{\ln a}\int du=\frac u{\ln a}+C=\frac{a^x}{\ln a}+C\]

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.1any questions about that?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0THANK YOU A MILLION TIMES OVER turning test! greatly appreciated
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