# Hot Integral - 12

**Calculus**Level 5

\[\displaystyle \int\limits_{-\infty}^{\infty} e^{-2\pi i \Psi x}f(x)dx =\frac{A-B\pi^C\Psi^D - E\pi^F\Psi^Gsgn(\Psi)}{H\pi^I\Psi^J}i - \frac{M\delta''(\Psi)}{K\pi^L}\]

where

\(\displaystyle f(x)=\frac{x^3 + x^2 + 2(1+xsgn(x))}{2x}\)

Calculate \[A+B+C+D+E+F+G+H+I+J+K+L+M\]

**Details and Assumptions**

\(\bullet \delta(\cdot)\) represents delta function.

\(\bullet i=\sqrt{-1}\)

\(\bullet sgn(x)=\frac{|x|}{x}\) represents sign function.

\(\bullet A,B,C,D,E,F,G,H,I,J,K,L,M\) are all positive integers.

\( \blacksquare \) This is a part of Hot Integrals

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