Concept preview

What you will be able to do

1. 01Apply the linearity of the definite integral, $\int_a^b\big(cf+dg\big)=c\int_a^b f+d\int_a^b g$, to combine constant multiples and sums of functions.
2. 02Use additivity over adjacent intervals, the reversed-bound sign rule, and the equal-bound rule to relate integrals over different intervals of the same function.
3. 03Compute a target definite integral from a table of given integral values, and avoid the reversed-bound sign error $\int_a^b f=-\int_b^a f$.

Property one is LINEARITY. Constants pull out of an integral and sums split apart: $\int_a^b\big(cf(x)+dg(x)\big)\,dx=c\int_a^b f(x)\,dx+d\int_a^b g(x)\,dx$. Geometrically, scaling a function by $c$ scales every signed-area sliver by $c$, and adding two functions stacks their heights so their areas add. This is what lets you break a complicated integrand into manageable pieces: $\int(3x^2-5\sin x)\,dx$ becomes $3\int x^2\,dx-5\int\sin x\,dx$. Linearity is the property you will lean on most often.