Struggling with Integration: e^x \sin(\pi x)

[ziddy83]

Ok, here is the integral i seem to be having some issues with. I know there's a very simple step I am missing.

$$\int_{}^{} e^x \sin(\pi x) dx$$

i attempted to do this using by parts integration.

I tried u = $$\sin(\pi x)$$ so du= $$\pi \cos(\pi x) dx$$
so then dv= $$e^x dx$$ and v= $$e^x$$

after using $$uv- \int v du$$ I seem to be going in circles...can someone help?

 

[quasar987]

Do integration by parts on $\int v du$. You'll end up with

$$\int_{}^{} e^x \sin(\pi x) dx = something - \int_{}^{} e^x \sin(\pi x) dx$$

Aaah.. add $\int_{}^{} e^x \sin(\pi x) dx$ on both sides. Divide by two. ta-dam.

 

[shmoe]

Divide by two.

Small warning, it won't look quite like you've described, the constant won't be two.

 

[quasar987]

Oh right.. because of the pies!

 

[Galileo]

There's a quick alternative way using complex exponentials.

Since
$$e^x \sin (\pi x)= \Im (e^{x+i\pi x})=\Im (e^{x(1+i\pi)})$$

$$\int e^{x(1+i\pi)} dx=\frac{e^{x(1+i\pi)}}{1+i\pi}=\frac{(1-i\pi)e^{x(1+i\pi)}}{1+\pi^2}$$

Now take the imaginary part.