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

1. TuringTest

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

2. virtus

a^x +c ?

3. virtus

actually i don't know hahaha

4. Mikey

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

5. TuringTest

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

6. virtus

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

7. TuringTest

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

8. virtus

9. TuringTest

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?

10. TuringTest

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

11. TuringTest

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$

12. TuringTest

any questions about that?

13. virtus

THANK YOU A MILLION TIMES OVER turning test! greatly appreciated

14. TuringTest

anytime :D