Question

evaluate the indefinite integral
integral of (6)/(e^(8x))dx

Answer 1

\[\int\limits \frac{ 6 }{ e ^{8x} }dx\]

Answer 2

can i use u substitution?

Answer 3

so if i make u=e^(8x) my du would be 1/8du = e^(8x)dx

Answer 4

so then i would have 1/8 integral of 6/u ?

Answer 5

not sure if i am right so far

Answer 6

is it possible to take out the u from the bottom?

Answer 7

is it true that if i take out u from the bottom of the fraction it would become u^(-1) ?

Answer 8

You're on the right track you'll need a substitution I think! I'll help walk you through this, first let's take out the coefficient 6 and use exponent rules to rewrite it like this:

\[6 \int e^{-8x}dx\]

From here you can use this substitution:
\[u=-8x \]

Take the derivative of this to get:

\[\frac{du}{dx} = -8\]
multiply \(dx\) by both sides:
\[du=-8dx\]
divide -8 from both sides:
\[\frac{-1}{8} du = dx\]

Now you can plug these into your for \(-8x=u\) and \(\frac{-1}{8}du = dx\)

\[6 \int e^u * \frac{-1}{8} du\]
Pull this constant out as well to get

\[\frac{-6}{8} \int e^udu\]
Now see if you can take it from here.

Answer 9

thank you i will try it

Answer 10

so the integral will be e^(u+1)/(u+1) ?

Answer 11

and i plug in u back in now which is -8x

Answer 12

so my answer is (e^(-8x+1))/((-8x+1)+1)

Answer 13

hmm i think i did it wrong that answer is not correct

Answer 14

So what you're doing is the rule for polynomials:

\[\int x^n dx = \frac{x^{n+1}}{n+1}\]

but the type of integral you're looking at is:
\[\int e^x dx = e^x\]

Which you can check by taking the deriativatives of the right hand sides of the equations since integrals are antiderivatives

Answer 15

oh i see, so the integral of e^u is just e^u... and all i would have to do is plug in my u and multiply it by -6/8 ...

Answer 16

so my answer is -3/4e^(-8x)

Answer 17

thank you!

Answer 18

Awesome glad I could help!