Originally Posted by
skeeter properties of definite integrals
1. $\displaystyle \int_6^{15} f(x) \, dx = \int_6^9 f(x) \, dx + \int_9^{12} f(x) \, dx + \int_{12}^{15} f(x) \, dx$
solve for $\displaystyle \int_9^{12} f(x) \, dx$ ... you were given the values for all the other definite integrals.
$\displaystyle \int_{12}^9 7f(x) - 7 \, dx = $
$\displaystyle 7\int_{12}^9 f(x) - 1 \, dx = -7\int_9^{12} f(x) - 1 \, dx = -7\int_9^{12} f(x) \, dx + 7\int_9^{12} 1 \, dx$
2. if $\displaystyle F(x) = \int_a^x f(t) dt$ , then $\displaystyle F'(x) = f(x)$