Question

Помогите плиз. Интеграл (х^2*е^(3х))

Answer 1

$\int x^2\cdot e^{3x}\, dx=[\, u=x^2\; ,\; du=2x\, dx,\; dv=e^{3x}\, dx,\; v=\frac{1}{3}e^{3x}\, ]=\\\\=\frac{1}{3}x^2e^{3x}-\frac{2}{3}\int x\, e^{3x}\, dx=[\, u=x\; ,du=dx,\; dv=e^{3x}dx,\; v=\frac{1}{3}e^{3x}\, ]=\\\\=\frac{1}{3}x^2e^{3x}-\frac{2}{3}\cdot (\frac{1}{3}xe^{3x}-\frac{1}{3}\int e^{3x}\, dx)=\\\\=\frac{2}{3}x^2e^{3x}-\frac{2}{9}xe^{3x}+\frac{2}{27}e^{3x}+C=\frac{2}{27}e^{3x}\cdot (9x^2-3x+1)+C$