Question

Integration Help ;

How to integrate combination of exponential and trignometric . If I use part by part integration method using u and v , exponential integration will outputs the same result . Help me....

Answer 1

It can be the same, it depends on which function you are given ...

Answer 2

can you post an example

Answer 3

Okay.....w8...

Answer 4

just keep doing it by parts until you get the same expression on the RHS as the one on the LHS so you will have something like F= blabla +blala2 + blabla3 + something*F. now just solve for F

Answer 5

where F is the original integral...

Answer 6

Please see my attached image....

Answer 7

is that e^-x or e^-n?

Answer 8

Its e^-x . Sorry for the bad handwriting .

Answer 9

let F= \[\int\limits_{0}^{1}e^{-x}\cos(n \pi x)dx\].

Now try doing integration by parts with u and v (for every integral use the same substitution for u and v ). You will end up with "something" + the original integral after few iterations. And if you just write F insted of your original integral you will have 2F=some_expression_here1 + some_expression_here2 + some_expression_here3 + some_constant*F

Answer 10

now just solve for F, since F = original integral

Answer 11

I will try to do a sketch and show you . Please check it .

Answer 12

I've done partial solution . I have done until 2 times part by part . Its just looking like gonna continue forever . Please check and advice .

Answer 13

ok, if you take out 1/pi*n in the very last integral you will get the original integral, that is F. Now, that whole row (last 1) is equal to F...now you have an algebraic equation, (solve for F)

Answer 14

i didn't check for any errors, i just checked the method...:D

Answer 15

Here's my solution hopefully there's no errors. I had a quick look. Good luck.

Answer 16

http://www.wolframalpha.com/input/?i=2+*integrate+from+0+to+1+%28e^%28-x%29+cos%28n*pi*x%29%29

Answer 17

In the fifth line, there shouldn't be a 2 infront of the n^2 pi^2 for my solution, the rest is good though.

Answer 18

Please check my image .

Answer 19

Okay now all you have to do is plug in the values

\[2 \int \limits_{0}^{1} e^{-x}cos{n\pi x} dx =\frac{ 2e^{-x}(n\pi sin{n\pi x} - cos{n\pi x})}{n^2\pi^2 +1}\]\[=\frac{ 2e^{-1}(n\pi sin{n\pi} - cos{n\pi })}{n^2\pi^2 +1}+\frac{ 2}{n^2\pi^2 +1}\]\[=\frac{ 2(n\pi sin{n\pi} - cos{n\pi })}{en^2\pi^2 +e}+\frac{ 2e}{en^2\pi^2 +e}\]\[=\frac{ 2(n\pi sin{n\pi} - cos{n\pi }+e)}{en^2\pi^2 +e}\]

Answer 20

okay yes I did bring that integral to the other side and added it, so let's look at just the numbers infront of the integral. you will have

\[1+\frac{1}{n^2\pi^2}=\frac{n^2\pi^2+1}{n^2\pi^2}\]

Answer 21

How you got 1 + 1/n^2.pi^2 . I got no idea how u got the 1 .

Answer 22

the one comes from the original integral.

\[\int e^{-x}cos{n\pi x} = ....... -\frac{1}{n^2\pi^2}\int e^{-x}cos{n\pi x}\]
\[1\int e^{-x}cos{n\pi x} +\frac{1}{n^2\pi^2}\int e^{-x}cos{n\pi x}= ....... \]
\[(1+\frac{1}{n^2\pi^2})\int e^{-x}cos{n\pi x}= ....... \]
\[(\frac{n^2\pi^2+1}{n^2\pi^2})\int e^{-x}cos{n\pi x}= ....... \]

Answer 23

Okay.....now i got u....thanks