## amyna one year ago solve the intergal:

1. amyna

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2. zepdrix

Oh this should be similar to the e^(-x^2) one that you had earlier :)

3. amyna

i got the answer: -1/2 ln10 * 10^-x^2 not sure if thats correct because i don't really understand this problem!

4. zepdrix

Your ln(10) is is the denominator with the 2? But the 10^(-x^2) is not, correct?

5. amyna

yes

6. zepdrix

Hmm ya that looks correct! :) What is confusing you on this one? :O Integration of the exponential base 10?

7. amyna

ya thats what i don't get, like where the ln 10 came from

8. zepdrix

Do you remember how to differentiate something like this? :)$\large\rm y=10^x$$\large\rm y'=?$

9. amyna

no i don't

10. zepdrix

$\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$

11. amistre64

works for base e as well

12. amistre64

y = e^x y' = e^x ln(e)

13. zepdrix

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.

14. amistre64

id rather learn one rule instead of special cases :)

15. zepdrix

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:$\large\rm \int\limits e^{\color{orangered}{a}x}dx=\frac{1}{\color{orangered}{a}}e^{\color{orangered}{a}x}$

16. zepdrix

Therefore,$\large\rm \int\limits 10^{x}dx=\int\limits e^{\color{orangered}{(\ln10)}x}dx=\frac{1}{\color{orangered}{\ln10}}e^{\color{orangered}{(\ln10)}x}=\frac{1}{\ln10}10^x$

17. amyna

Thank You so much! It make so much more sense now! :)

18. zepdrix

Ah I made a typo halfway through :( 10^(x(ln10)) should be e^(x(ln10))

19. zepdrix

Could you make sense of some of that? :O I know it was a lot to take in all at once lol

20. amyna

haha yes! Thank You!