Question

integrate y=2^x from x=0 to x=3

Answer 1

\[\frac d{dx}a^x=a^x\ln a\]so then what is the integral?

Answer 2

a^x +c ?

Answer 3

actually i don't know hahaha

Answer 4

The indefinite integral of a^x is a^x/lna + C

Answer 5

\[\int a^x\ln adx=a^x\]so\[\int a^xdx=\frac{a^x}{\ln a}\]+C if it is indefinite, but yours is definite anyway...

Answer 6

turning test can you please show me how this was derived. Thank you very much!

Answer 7

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

Answer 8

yes please!

Answer 9

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?

Answer 10

\[y'=a^x\ln a\iff y=a^x\]so\[\int a^x\ln a=a^x\]now we can use this info and do your integral with a u-substitution

Answer 11

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\[\frac1{\ln a}\int du=\frac u{\ln a}+C=\frac{a^x}{\ln a}+C\]

Answer 12

any questions about that?

Answer 13

THANK YOU A MILLION TIMES OVER turning test! greatly appreciated

Answer 14

anytime :D