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solve the intergal:

Mathematics
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|dw:1441503440519:dw|
Oh this should be similar to the e^(-x^2) one that you had earlier :)
i got the answer: -1/2 ln10 * 10^-x^2 not sure if thats correct because i don't really understand this problem!

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Your ln(10) is is the denominator with the 2? But the 10^(-x^2) is not, correct?
yes
Hmm ya that looks correct! :) What is confusing you on this one? :O Integration of the exponential base 10?
ya thats what i don't get, like where the ln 10 came from
Do you remember how to differentiate something like this? :)\[\large\rm y=10^x\]\[\large\rm y'=?\]
no i don't
\[\large\rm y=10^x\]\[\large\rm \ln y=\ln 10^x\]\[\large\rm \ln y=x \ln 10\]Differentiating,\[\large\rm \frac{1}{y}y'=\ln 10\]\[\large\rm y'=y (\ln 10)\]\[\large\rm y'=10^x (\ln 10)\]So whenever we differentiate an exponential of base NOT e, we multiply by natural log of the base. When we integrate, we do the inverse of that, we divide by natural log of the base.\[\large\rm \int\limits 10^x dx=\frac{1}{\ln10}10^x\]
works for base e as well
y = e^x y' = e^x ln(e)
Yes, but unnecessary :) I guess that's why they don't show it for the derivative of e^x. I like seeing that extra step in the middle, helps to see the rule without forgetting about it.
id rather learn one rule instead of special cases :)
You could also do this maybe Amy :O Recall that since the exponential base e, and natural log are inverse operations,\[\large\rm 10^x=e^{(\ln10^x)}\]Applying exponent rule gives,\[\large\rm =10^{x\color{orangered}{(\ln10)}}\]And maybe you remember how to integrate something of this form:\[\large\rm \int\limits e^{\color{orangered}{a}x}dx=\frac{1}{\color{orangered}{a}}e^{\color{orangered}{a}x}\]
Therefore,\[\large\rm \int\limits 10^{x}dx=\int\limits e^{\color{orangered}{(\ln10)}x}dx=\frac{1}{\color{orangered}{\ln10}}e^{\color{orangered}{(\ln10)}x}=\frac{1}{\ln10}10^x\]
Thank You so much! It make so much more sense now! :)
Ah I made a typo halfway through :( 10^(x(ln10)) should be e^(x(ln10))
Could you make sense of some of that? :O I know it was a lot to take in all at once lol
haha yes! Thank You!

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