定積分の公式：定数倍、足し算と引き算

定数倍の定積分

\[ \int_a^{b} \! cf(x) dx = c \int_a^{b} \! f(x) dx \]

定積分の足し算と引き算

\[ \int_a^{b} \! (f(x) + g(x)) dx = \int_a^{b} \! f(x) dx + \int_a^{b} \! g(x) dx \\ \\ \int_a^{b} \! (f(x) - g(x)) dx = \int_a^{b} \! f(x) dx - \int_a^{b} \! g(x) dx \]

両端の交換

\[ \int_b^{a} \! f(x) dx = -\int_a^{b} \! f(x) dx \]

端の連結

\[ \int_a^{b} \! f(x) dx + \int_b^{c} \! f(x) dx = \int_a^{c} \! f(x) dx \]

定積分の公式を使った例題

\begin{split} \int_a^{b} \! -4x dx &= -4 \int_a^{b} \! x dx \end{split}

\begin{split} \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 \end{split}