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Frequently Used Definite Integrals of Quadratic Functions 📂Lemmas

Frequently Used Definite Integrals of Quadratic Functions

Formula

∫αβ(x−α)(x−β)dx=−(β−α)36 \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

∫αβ(x−α)(x−β)dx=∫αβx2−(α+β)x+αβdx=β3−α33−(α+β)β2−α22+αβ(β−α)=2β3−2α3−3β3−3αβ2+3α2β+3α3+6αβ2−6α2β6=2β3−3β3+3α3−2α3+6αβ2−3αβ2+3α2β−6α2β6=−β3+α3+3αβ2+−3α2β6=−(β−α)36 \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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