Question

Evaluate the integral: e^2x-1/e^x dx

Answer 1

put e^-x =t

Answer 2

if you have only just started learning this stuff, do them separately and make substitutions - which is probs what they are trying to show you.

for the first part, go u = 2x, du = 2dx: 2∫exp(u) du
for the second, go, u = -x, du = -dx: -∫exp(u) du

in due course, you should just be doing this in your head.

Answer 3

or separate the terms

Answer 4

Its actually a definite integral. Can I still use u substitution for definite integrals?

Answer 5

yep

Answer 6

just make sure you cnahge the interval to one that reflects u. or convert the integral back to x and use the orig intervals.

Answer 7

I just have trouble with e^x problems. Is there any way you can draw this?

Answer 8

certainly get used to drawing y = e^x, y = e^(-x) and y = ln(x). these are really really important in maths; and nature, because maths is how we model nature. but geometry/ drawing does not help IMHO.

just remember this stuff:

d/dx(e^x) = e^x
∫e^x dx = e^x
THIS IS WHY e IS SO IMPORTANT; and this is how we calculate the value of e in the first place, ie to get this relationship.

and also
∫ 1/x dx = ln(x)
d/dx(ln(x) = 1/x

these last 2 actually follow straight from the first but it does no harm to remember them.

SO much follows from this. whether via substitution or whatever.