定積分の公式：定数倍、足し算と引き算

定数倍の定積分

$\int_{a}^{b} c f (x) d x = c \int_{a}^{b} f (x) d x$

定積分の足し算と引き算

$\int_{a}^{b} (f (x) + g (x)) d x = \int_{a}^{b} f (x) d x + \int_{a}^{b} g (x) d x \int_{a}^{b} (f (x) - g (x)) d x = \int_{a}^{b} f (x) d x - \int_{a}^{b} g (x) d x$

両端の交換

$\int_{b}^{a} f (x) d x = - \int_{a}^{b} f (x) d x$

端の連結

$\int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x = \int_{a}^{c} f (x) d x$

定積分の公式を使った例題

$\int_a^{b} \! -4x dx &= -4 \int_a^{b} \! x dx$

$\int_a^{b} \! (3x^2 - 4x + 5) dx &= \int_a^{b} \! 3x^2 dx + \int_a^{b} \! (-4x) dx + \int_a^{b} \! 5 dx \\ &= 3 \int_a^{b} \! x^2 dx - 4 \int_a^{b} \! x dx + 5 \int_a^{b} \! 1 dx$