Integration by parts?

Question

I tried this question but it seemed that both the cos x and e^x wont disappear.
how do you integrate:
e^x * cos(x) dx

I tried this question but it seemed that both the cos x and e^x wont disappear.

please show the steps
thanks!

Answer 1

You have to do 2 integrations by parts and your initial integral reappears with a minus sign. Then you are done.

Answer 2

You are right its a two part integration by parts
So separate into Cosx and e^x
use parts and you get e^xcosX plus integral of e^xsinx
Then Keep going separate e^x and sinX
So integrating by parts the second bit you get e^xsinx minus the integral you first wanted
So call this integral J.
J is therefore e^xcosx plus e^xsinx minus J
Hence 2J is e^xcosx plus e^xsinx
So J is (1/2)*(e^x*cos(x)+e^x*sin(x)) oh plus a constant of course.

Differentiate it back by product rule and it is seen to be correct
Make sense?

Answer 3

∫e^x * cos(x) dx
let u = e^x, du = e^xdx
knowing
∫udv = uv - ∫vdu

dv = cos(x)dx, v = + sin(x)

∫e^x * cos(x) dx = + e^xsin(x) + ∫e^xsin(x)dx
for the last onel
let u = e^x, du = e^xdx
dv = sin(x)dx, v =- cos(x)

∫e^xcos(x)dx = e^xsin(x) + e^xcos(x) - ∫e^xcos(x)dx

2∫e^xcos(x)dx = e^xcos x + e^xsin(x)
or ∫e^xcos(x)dx = e^x(cos x + sin x)/2
u should add a constant of integration to get
∫e^xcos(x)dx = e^x(cos x + sin x)/2 + C

Answer 4

take I= integtation of e^x* cos(x) dx
after solving once or twice u will get this termtake all I's on one side and u will get the answer

Answer 5

Helmut your integral is wrong
Astrape is correct
Nathan

Answer 6

I think you have to do it twice:

See wikipedia article "integration by parts"

Answer 7

In this case you don't want e^x and cos(x) to disappear:
∫e^x * cos(x) dx
let u = e^x, du = e^xdx
dv = cos(x)dx, v = - sin(x)
∫e^x * cos(x) dx = - e^xsin(x) + ∫e^xsin(x)dx
let u = e^x, du = e^xdx
dv = sin(x)dx, v = cos(x)
∫e^xcos(x)dx = - e^xsin(x) + e^xcos(x) - ∫e^xcos(x)dx
2∫e^xcos(x)dx = e^xcos(x) - e^xsin(x)
∫e^xcos(x)dx = ((e^x)/2)(cos(x) - sin(x)) + C

Answer 8

I = ∫ cos x.e^x.dx
∫ u .(dv/dx).dx = uv - ∫ v.(du/dx).dx
where u = cos x and dv/dx = e^x
du/dx = -sin x
v = e^x
I = cos x.e^x + ∫ e^x.sin x .dx
Ia = ∫ e^x.sin x.dx
Ia = sin x.e^x - ∫ e^x.cos x.dx
Ia = sin x.e^x - I
I = cos x.e^x + sin x.e^x - I + C
2I = cos x.e^x + sin x.e^x + C
I = (1/2).cos x.e^x + (1/2).sin x.e^x + K