exponential function with polynomial exponent

Question

hi guys can anyone help me, I am currently working with some integrals and i am tangled with $$\int_0^1 x^2(1-x)e^{-ax^2+b(1-x)^2}dx$$ and $$\int_0^\infty x^{-1}e^{-ax^2-bx^{-2}}dx$$ I had tried completing the square and substitution

Answer 1

Set $t=a-b$. Then your integral is $$ \int_0^1 x^2(1-x)e^{-ax^2+b(1-x)^2}dx=-\frac{\partial}{\partial t}\int_0^1 dx\ e^{-t x^2-2 b x+b}+\frac{e^{b}}{8}\frac{\partial^3}{\partial b^3}\int_0^1 dx\ e^{-t x^2-2 b x}\ , $$ evaluated at $t=a-b$ at the end of the calculation. Then complete the square and use the definition of the error function (Erf) https://en.wikipedia.org/wiki/Error_function

The final result is (assuming $a>b$) $$ \frac{1}{{4 (a-b)^{7/2}}}e^{-a} \left(\sqrt{\pi } \left((2 b+3) b (a-b)+(a-b)^2+2 b^3\right) e^{\frac{a^2}{a-b}} \left(\text{erf}\left(\frac{a}{\sqrt{a-b}}\right)-\text{erf}\left(\frac{b}{\sqrt{a-b}}\right)\right)+2 \sqrt{a-b} \left(-(a b+a-b) e^{a+b}+a+b^2-b\right)\right)\ . $$