Integrating with the number e

[jonnburton]

Hi everyone,

I'm having trouble applying integration to equations which include the number e. The following is an example of a question which I can't seem to get right. Would anybody be able to tell me what the correct way to approach this is?

Evaluate:

\(\displaystyle \int^1_0(4x-5e^{-x})\,dx\)

\(\displaystyle \int(4x-5e^{-x})\,dx\) = \(\displaystyle 4 + 5e^{-x}\)

\(\displaystyle \left[4+5e^{-x}\right]_0^1 = \left[4+5e^{-1}\right] - \left[4+5e^{0}\right]\)

= \(\displaystyle 4 + 5e^{-1} -9\)

However, the correct answer according to the book is \(\displaystyle 5e^{-1} - 3\) which doesn't make any sense to me..!

Many thanks in advance for any advice on this!

 

[DrPhil]

Hi everyone,

I'm having trouble applying integration to equations which include the number e. The following is an example of a question which I can't seem to get right. Would anybody be able to tell me what the correct way to approach this is?

Evaluate:

\(\displaystyle \int^1_0(4x-5e^{-x})\,dx\)

\(\displaystyle \int(4x-5e^{-x})\,dx\) = \(\displaystyle 4 + 5e^{-x}\)......X

\(\displaystyle \left[4+5e^{-x}\right]_0^1 = \left[4+5e^{-1}\right] - \left[4+5e^{0}\right]\)

= \(\displaystyle 4 + 5e^{-1} -9\)

However, the correct answer according to the book is \(\displaystyle 5e^{-1} - 3\) which doesn't make any sense to me..!

Many thanks in advance for any advice on this!

The exponential happens to be correct - your error lies in \(\displaystyle \int 4x\ dx \). It looks like you took the derivative instead of the anti-derivative! [Actually, that would have been ok for the exponential .. either integration of differentiation of e^-x gives the same.]

\(\displaystyle \int 4x\ dx = 2 x^2\) will fix everything.

 

[jonnburton]

Thanks a lot DrPhil. Yes, I had mistakenly taken the anti-derivative rather than the derivative there.

I've got to the answer now - thanks again for clearing that up!