Four properties for taking integrals apart. A constant factor pulls outside the integral; a sum or difference is integrated term by term; a definite integral splits at any interior point of its interval. The one move that is not allowed: integrating a product as the product of the separate integrals.
A handful of properties let you take a complicated integral apart into pieces you already know how to handle. Three of them split an integral up; one common-looking move is not allowed at all.
A constant factor can be lifted out in front of the integral, left there while you integrate what remains, and multiplied back in at the end. It never interferes with the integration itself.
The added constant absorbs it: times is still just an unknown constant, so it is written again.
Take the factor outside, integrate the rest, then multiply back.
In every case the constant waits outside until the integration is done.
An integral is the area between the curve and the -axis. Multiplying the function by a constant stretches that area by the same factor: doubles it, a factor between and shrinks it, and a negative factor flips it to the other side of the axis.
Integrate each term on its own and combine the results with the sign that joined them. Nothing needs to be rearranged first.
Each term brings its own , but the sum of unknown constants is a single unknown constant, so only one is written.
As areas: adding two functions stacks their heights, so the region under is the region under plus the region under .
A definite integral over an interval equals the sum of the integrals over its pieces. Pick any point inside and cut there.
The rules above cover a constant times a function, and a sum or difference of functions. A product of two functions is where the pattern stops.
A quick check shows why: , but multiplying the separate integrals gives — not the same expression.
Integrating a product needs its own method, covered in a later lesson.