Frequently Used Definite Integrals of Quadratic Functions
Formula
$$ \int _{ \alpha }^{ \beta }{ (x-\alpha )(x-\beta )dx }=-\frac { { (\beta -\alpha ) } ^ { 3 } }{ 6 } $$
Description
As you solve problems, you often find yourself calculating definite integrals of this sort more than you’d expect. This formula is entirely useless aside from making solutions quicker, and its derivation is just calculation. Just memorize the form so you can use it instantly.
Derivation
$$ \begin{align*} & \int _{ \alpha }^{ \beta }{ (x-\alpha )(x-\beta )dx } \\ =& \int _{ \alpha }^{ \beta }{ { {x }^2-(\alpha +\beta )x+\alpha \beta }dx } \\ =& \frac { \beta^3-{ \alpha^3 } }{ 3 }-(\alpha +\beta )\frac { \beta^2-\alpha^2}{ 2 }+\alpha \beta (\beta -\alpha ) \\ =& \frac { 2\beta^3-2\alpha^3-3\beta^3-3\alpha \beta^2+3\alpha^2\beta +3\alpha^3+6\alpha \beta^2-6\alpha^2\beta }{ 6 } \\ =& \frac { 2\beta^3-3\beta^3+3\alpha^3-2\alpha^3+6\alpha \beta^2-3\alpha \beta^2+3\alpha^2\beta -6\alpha^2\beta }{ 6 } \\ =& \frac { - \beta^3 + \alpha^3 + 3\alpha \beta^2 + - 3 \alpha^2\beta }{ 6 } \\ &=-\frac { { (\beta -\alpha ) } ^ { 3 } }{ 6 } \end{align*} $$
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