# Need help solving integral-$\int x^2 (e^x)^3 \sin x \mathrm dx$

i have a nasty integral to solve. i tried using the parts method but it just blew off.Can anyone provide a solution much better than what i have done here?.

$$\int x^2 (e^x)^3 \sin x\, \mathrm dx$$

• $(e^x)^3=e^xe^xe^x=e^{3x}$ – Adi Dani Apr 30 '13 at 18:11

Just integrate by parts: $$\int x^2\left(e^x\right)^3\sin x\,\mathrm{d}x=\int u\,\mathrm{d}v\text{ where }u=x^2,\,\mathrm{d}v=e^{3x}\sin x\,\mathrm{d}x$$
Note you may want to tackle $v=\int e^{3x}\sin x\,\mathrm{d}x$ using parts as well, or you could (of course) play with Euler's identity.
Here is another approach. Put $$\Phi(t) = \int e^{3tx}\sin(x)\, dx.$$ Now resolve the integral by integration by parts.
Next, apply Leibnitz's rule for differentiation of the integral to find $\Phi''(t)$ and you are on home base.