find the integral
5^-x dx

Question

find the integral
5^-x dx

zepdrix · Answer

$\Large\bf \color{#CC0033}{\text{Welcome to OpenStudy! :)}}$

zepdrix · Answer

$$\Large\bf\sf \int\limits 5^{-x}\;dx$$

zepdrix · Answer

Do you remember the derivative of this?$$\Large\bf\sf 5^x$$

anonymous · Answer

is with natural log???

zepdrix · Answer

Yes :o

rvc · Answer

5^-x/log5 +c is this correct? @zepdrix

zepdrix · Answer

You weren't the person who asked the question...

rvc · Answer

i know so i can't ans

anonymous · Answer

so is the derivative of 5^x just ln5???

rvc · Answer

@zepdrix

anonymous · Answer

and if it were negative would it be -ln5?

zepdrix · Answer

$$\Large\bf\sf (5^x)'\quad=\quad 5^x \ln 5$$

rvc · Answer

yes

zepdrix · Answer

With exponentials, you get the same thing back, but with a factor of log of the base.

zepdrix · Answer

So when we integrate, instead of multiplying by that factor of ln5, we'll divide by it.

rvc · Answer

general formula

anonymous · Answer

so ln5/5^x?

zepdrix · Answer

Same thing with the chain rule.
Instead of getting an extra -1 from the chain rule, we'll instead divide by a -1.

If you want we can be more formal about it though :)
Rewrite our integral like this,$$\Large\bf\sf \int\limits 5^{-x}\;dx\quad=\quad \int\limits e^{\log(5^{-x})}\;dx$$And then we can do a u-sub.

zepdrix · Answer

Yah looks like you've got the right idea, let's just deal with the negative as well :U

rvc · Answer

what about me @zepdrix 
u didnot ans me

anonymous · Answer

ooh so -ln5/5^x or no????

zepdrix · Answer

Yah that looks good \c:/

zepdrix · Answer

@rvc I just wish you would help instead of `posting a direct answer`.
That's really frustrating.

rvc · Answer

but was i correct? @zepdrix

anonymous · Answer

and now u substitution? what would i set u equal to??? 5^x??

zepdrix · Answer

Hmm I think you missed a negative rvc :(

zepdrix · Answer

No, the u-sub is a different method for solving the problem.
It takes a little longer, but it's a good practice method if you're not totally comfortable with integrals yet.

We don't need to u-sub, we already solved it.

rvc · Answer

yes its -ve

rvc · Answer

so my ans was not direct understood

rvc · Answer

@zepdrix

anonymous · Answer

oh that wasnt too bad! alrighty thank you!!!

zepdrix · Answer

np \c:/