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1. I want to integrate e^x.cos(nx) (where n is number)

I use the integration by parts method where uv - Int(v.du/dx)

u = cos(nx) dv/dx = e^x
du/dx = -nsin(nx) v = e^x

Therefore

= cos(nx).e^x - Int(-nsin(nx).e^x)
= cos(nx).e^x + n.Int(sin(nx).e^x)

I have to repeat this same method again for "Int(sin(nx).e^x)"

u = sin(nx) dv/dx = e^x
du/dx = ncos(nx) v = e^x

So now:
cos(nx).e^x + n [sin(nx).e^x - n.Int(cos(nx).e^x)]
Using I = Int[e^x.cos(nx)]
cos(nx).e^x + n [sin(nx).e^x - n.I]
cos(nx).e^x + nsin(nx).e^x - n^2.I
I.n^2 = cos(nx).e^x + nsin(nx).e^x
I = [e^x (cos(nx) + nsin(nx))]/n^2

Im sorry if thats hard to understand
but ive also attached the handwritten version

I checked my answer on Wolphram alpha,
they have the same thing except in the denominator they have
n^2 +1 .. where as I have n^2 only

If someone could point out the mistake I've made - I would really appreciate it!
Attached Images

2. (Original post by Sifr)
I want to integrate e^x.cos(nx) (where n is number)

I use the integration by parts method where uv - Int(v.du/dx)

u = cos(nx) dv/dx = e^x
du/dx = -nsin(nx) v = e^x

Therefore

= cos(nx).e^x - Int(-nsin(nx).e^x)
= cos(nx).e^x + n.Int(sin(nx).e^x)

I have to repeat this same method again for "Int(sin(nx).e^x)"

u = sin(nx) dv/dx = e^x
du/dx = ncos(nx) v = e^x

So now:
cos(nx).e^x + n [sin(nx).e^x - n.Int(cos(nx).e^x)]
Using I = Int[e^x.cos(nx)]
cos(nx).e^x + n [sin(nx).e^x - n.I]
cos(nx).e^x + nsin(nx).e^x - n^2.I
I.n^2 = cos(nx).e^x + nsin(nx).e^x
I = [e^x (cos(nx) + nsin(nx))]/n^2

Im sorry if thats hard to understand
but ive also attached the handwritten version

I checked my answer on Wolphram alpha,
they have the same thing except in the denominator they have
n^2 +1 .. where as I have n^2 only

If someone could point out the mistake I've made - I would really appreciate it!
You have , adding to both sides gives where the LHS factorises to ; not i.e. your mistake is two lines from the bottom of your written working.

Unrelated, but your handwriting is excellent!
3. Wow, agreed, your handwriting is very nice

On the last line, did you draw the fraction with a ruler or by hand 0.o
4. (Original post by Farhan.Hanif93)
You have , adding to both sides gives where the LHS factorises to ; not i.e. your mistake is two lines from the bottom of your written working.

Unrelated, but your handwriting is excellent!
The fact that I have good handwriting just proves why I am a terrible mathematician!
5. (Original post by Sifr)
The fact that I have good handwriting just proves why I am a terrible mathematician!
It also means you'll never become a doctor

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