# Hot Integral - 17

Calculus Level 5

$\large \int\limits_{-\infty}^{\infty} (f(x) + g(x))e^{-2\pi i \lambda x} dx = \Re - i\Im$

where

$$\displaystyle f(x)=\frac{1+2x^2(\sin(x^2)+\log(|x|))}{2x}$$

$$\displaystyle g(x)=\frac{sgn^2(x-\frac{1}{2})+sgn^2(x+\frac{1}{2})}{4}-\frac{sgn(x-\frac{1}{2})sgn(x+\frac{1}{2})}{2}$$

$\displaystyle \Re = \frac{\sin^a(\lambda \pi)}{(\lambda^b\pi^c)} + \frac{d}{e\lambda^f|\lambda|} - \gamma\delta^{(h)}(\lambda)$

$\displaystyle \Im = \pi^j(\lambda^k\pi^{\frac{l}{m}}\cos((\lambda\pi)^n - \frac{\pi^p}{q}) + \frac{r}{s}sgn(\frac{\lambda}{t}))$

Find $$a+b+c+d+e+f+h+j+k+l+m+n+p+q-r+s+t$$

Clarifications:

• $$\delta^{(n)}(\cdot)$$ represents the $$n^{th}$$ derivative of delta function.

• $$i=\sqrt{-1}$$

• $$|\cdot|$$ represents modulus function.

• $$sgn(x)=\frac{x}{|x|}$$

• $$\Re , \Im$$ means real and imaginary part respectively.

• $$\gamma$$ is Euler -Mascheroni constant.

This is a part of Hot Intergals

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