常见积分

$$integral cos^2 = 1 / 2 x + 1 / 4 sin(2x) + C$$

验证 $prime = 1 / 2 + 1 / 2 cos (2x) = 1 / 2 (1 + c^2 - s^2) = c^2$

$$ integral e^(-x ^ 2) = sqrt(pi) \ integral e^(-x^2)x^2 = sqrt(pi) / 2 $$

ref: https://math.stackexchange.com/questions/66084/integral-int-infty-infty-x2-e-x2-mathrm-dx