Question

integrate Sqrt(1+e^x) ?

Answer 1

multiply both top and bottom by e^x, then let u = e^x

Answer 2

do that and what do you get?

Answer 3

im not sure!

Answer 4

i dont understand how that helps!

Answer 5

sqrt(1+e^x) e^x dx / e^x

u = e^x,
du = e^x dx

sqrt(1 + u)/u du

yes?

Answer 6

yeah that makes sense

Answer 7

now let t = sqrt(1 + u), what do you get?

Answer 8

t/u du?

Answer 9

solve for u

Answer 10

I'm sorry after spring break i feel so dumb, i'm not sure

Answer 11

is this some what on the right path?

Answer 12

you could also try making \(u=1+e^x\)

Answer 13

t = sqrt(1 + u)
t^2 = 1 + u
u = t^2 - 1
du = 2t dt

t (2t) / (t^2 - 1) dt

2t^2 / (t^2 - 1)

The integral now turns to partial composition type

Answer 14

another way to skin the cat, go the whole hog and make
\[y=\sqrt{1+e^x}\]

Answer 15

you get
\[u=\sqrt{1+e^x}\]
\[u^2=1+e^x\]
\[u^2-1=e^x\]
\[\ln(u^2-1)=x\]
\[dx=\frac{1}{u-1}+\frac{1}{u+1}\]

Answer 16

so to get from 2t^2/t^2-1 dt to the last step you posted I just use partial fractions?
@sourwing