## virtus Group Title integrate y=2^x from x=0 to x=3 2 years ago 2 years ago

1. TuringTest Group Title

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

2. virtus Group Title

a^x +c ?

3. virtus Group Title

actually i don't know hahaha

4. Mikey Group Title

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

5. TuringTest Group Title

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

6. virtus Group Title

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

7. TuringTest Group Title

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 Group Title

9. TuringTest Group Title

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 Group Title

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

11. TuringTest Group Title

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 Group Title