Integral of a standard gaussian distribution

Question

Please see the image. I know that the integral of P(x) tends to 1. But the quadratic equation next to P(x) seems confusing.

Answer 1

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You are trying to evaluate: $$ \int p(x) (ax^2 + bx+c)dx $$ you can break this up into: $$ a \int p(x)x^2dx + b\int p(x)xdx+ c\int p(x)dx $$ Then use the following facts: $$ \int p(x) x^2dx = \mathbb{E}(x^2) = Var(x) + \mathbb{E}(x)^2 $$ and $$ \int p(x) x dx= \mathbb{E}(x) $$ and finally: $$ \int p(x)dx = 1 $$