Question

help please ...
evaluate the integral

Answer 1

\[\int\limits_{}^{} e^x \cos (x) dx \]

Answer 2

okay so i did this

u= cos (x)
du = - sin (x)


dv = e^x
v= e^x



cos (x) e^x - \[\int\limits_{}^{}e^x *- \sin (x)\]

Answer 3

@zepdrix

Answer 4

@agent0smith

Answer 5

@Kainui

Answer 6

you need to apply integration by parts TWICE here. So do it again for e^x sin(x)

Answer 7

It will give you a nice recursion of the original integral.

Answer 8

okay so
u= e^x
du = e^x

dv= cos (5x)
v= sin (5x)/ 5

Answer 9

This is tricky, you've done it right so far.

\[\int e^x *\cos (x)dx = \cos (x) e^x - \int e^x *- \sin (x)dx\]

Now the trick is you need to do is repeat this AGAIN:

u= sin(x)
du = cos (x)dx

dv = e^x
v= e^x

and when you plug it in you'll end up with the integral on both sides of the equation, it's strange but then you can solve for your integral.

Answer 10

ah yes thats where it confuses me

Answer 11

aahh have no idea what i did wrong .. tried it like 3 times

Answer 12

\[I = \int\limits e^x \cos(x) dx = e^x \cos(x) + \int\limits e^x \sin(x) \]
\[\int\limits e^x \sin(x) dx = e^x \sin (x) - \int\limits e^x \cos(x) dx = e^x \sin(x) - I\]

\[I = e^x \cos(x) + e^x \sin(x) - I \rightarrow 2I = e^x (\cos(x) + \sin(x))\]

Answer 13

ohhh i see what i did wrong lol

Answer 14

okay got it ! hmm i have a quiz on this today .. whats the important thing that should look into more ?

Answer 15

I'd say practice on figuring out the best part to integrate at the start.

Generally the rule is ETALE - exponentials, trig, algebraic (including 1), logs and then everything else.
The exercise you just did shows you a property of some IBP integrals where it produces a recurring integral, and that will be covered in topics such as reduction formula.
Also get used to doing some of the definite integrals as well.

Answer 16

@marcelie how'd you end up with
dv= cos (5x)
v= sin (5x)/ 5

where is the 5 coming from?

Answer 17

hmm isnt that the integral ?

Answer 18

No... where did the 5 come from??