Tricky Integral

[goku900]

I need help integrating this definite integral

0 to pi ∫ (e^cos(x))sin(2x) dx

So far I have used a double double angle identity where sin2x=2sin(x)cos(x)

I have ∫e^cos(x) * 2sin(x)cos(x) dx

I let u=cos(x)
du=-sinx dx
dx=-du/sinx

This turns to

- ∫[(e^u) (2u)(sinx)]/sinx du
-2 ∫u(e^u) du <==== stuck here, don't even know if I'm doing this right so far.

 

[Deleted member 4993]

I need help integrating this definite integral

0 to pi ∫ (e^cos(x))sin(2x) dx

So far I have used a double double angle identity where sin2x=2sin(x)cos(x)

I have ∫e^cos(x) * 2sin(x)cos(x) dx

I let u=cos(x)
du=-sinx dx
dx=-du/sinx

This turns to

- ∫[(e^u) (2u)(sinx)]/sinx du
-2∫u(e^u) du <==== stuck here, don't even know if I'm doing this right so far.

Looks Okay upto here....

Now use integration by parts - that states:

\(\displaystyle \displaystyle \int f(u) . d[g(u)] \ = \ f(u) * g(u) \ - \ \int g(u) d[f(u)]\)

in your case:

f(u) = u

and

d[g(u)] = e^udu .................. continue

 

[goku900]

Ok thanks. I did the by parts before posting but I wasn't sure so I didn't post it originally

So after doing the by parts I got

= -2(ue^(u) - e^(u) eval. from 1 to -1)

I'm having a brain malfunction and I'm not seeing how those numbers plugin. If I could just see how this is done out that'd be appreciated.

I'm supposed to get the answer 4/e but idk how.

 

[DrPhil]

Ok thanks. I did the by parts before posting but I wasn't sure so I didn't post it originally

So after doing the by parts I got

= -2(ue^(u) - e^(u) eval. from 1 to -1)

I'm having a brain malfunction and I'm not seeing how those numbers plugin. If I could just see how this is done out that'd be appreciated.

I'm supposed to get the answer 4/e but idk how.

When u=+1, the two terms cancel.

When u=-1, then you get

\(\displaystyle (-2)\left[ (-1)e^{-1} - e^{-1}\right] =\ . . .\)

 

[Deleted member 4993]

Ok thanks. I did the by parts before posting but I wasn't sure so I didn't post it originally

So after doing the by parts I got

= -2(ue^(u) - (-2)e^(u) eval. from 1 to -1)

I'm having a brain malfunction and I'm not seeing how those numbers plugin. If I could just see how this is done out that'd be appreciated.

I'm supposed to get the answer 4/e but idk how.

-2 * [ {(1)*e¹ - e¹} - {(-1)*e^-1 - e^-1}]

and continue.....