# Primitive of x squared by Root of a x squared plus b x plus c

Jump to navigation Jump to search

## Theorem

Let $a \in \R_{\ne 0}$.

Then:

$\displaystyle \int x^2 \sqrt {a x^2 + b x + c} \rd x = \frac {6 a x - 5 b} {24 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {5 b^2 - 4 a c} {16 a^2} \int \sqrt {a x^2 + b x + c} \rd x$

## Proof

 $\ds$  $\ds \int x^2 \sqrt {a x^2 + b x + c} \rd x$ $\ds$ $=$ $\ds \int \frac {2 a x^2} {2 a} \sqrt {a x^2 + b x + c} \rd x$ multiplying top and bottom by $2 a$ $\ds$ $=$ $\ds \int \frac {x \paren {2 a x + b - b} } {2 a} \sqrt {a x^2 + b x + c} \rd x$ $\text {(1)}: \quad$ $\ds$ $=$ $\ds \frac 1 {2 a} \int x \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x - \frac b {2 a} \int x \sqrt {a x^2 + b x + c} \rd x$ Linear Combination of Integrals

Let:

 $\ds z$ $=$ $\ds a x^2 + b x + c$ $\ds \leadsto \ \$ $\ds \frac {\d z} {\rd x}$ $=$ $\ds 2 a x + b$ Derivative of Power $\ds \leadsto \ \$ $\ds \int \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x$ $=$ $\ds \int \sqrt z \rd z$ Integration by Substitution $\ds$ $=$ $\ds \frac {2 \paren {\sqrt z}^3} 3$ Primitive of Power $\text {(2)}: \quad$ $\ds$ $=$ $\ds \frac {2 \paren {\sqrt {a x^2 + b x + c} }^3} 3$ substituting for $z$

With a view to expressing the primitive $\displaystyle \int x \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x$ in the form:

$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

 $\ds u$ $=$ $\ds x$ $\ds \leadsto \ \$ $\ds \frac {\d u} {\d x}$ $=$ $\ds 1$ Derivative of Identity Function

and let:

 $\ds \frac {\d v} {\d x}$ $=$ $\ds \int \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x$ $\ds \leadsto \ \$ $\ds v$ $=$ $\ds \frac {2 \paren {\sqrt {a x^2 + b x + c} }^3} 3$ from $(2)$ above

Then:

 $\ds$  $\ds \int x \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x$ $\ds$ $=$ $\ds x \paren {\frac {2 \paren {\sqrt {a x^2 + b x + c} }^3} 3} - \int \paren {\frac {2 \paren {\sqrt {a x^2 + b x + c} }^3} 3} \times 1 \rd x + C$ Integration by Parts $\text {(3)}: \quad$ $\ds$ $=$ $\ds \frac {2 x \paren {\sqrt {a x^2 + b x + c} }^3} 3 - \frac 2 3 \int \paren {\sqrt {a x^2 + b x + c} }^3 \rd x + C$ simplifying

Now consider:

 $\ds$  $\ds \int \paren {\sqrt {a x^2 + b x + c} }^3 \rd x$ $\ds$ $=$ $\ds \int \paren {a x^2 + b x + c} \sqrt {a x^2 + b x + c} \rd x$ factorising $\text {(4)}: \quad$ $\ds$ $=$ $\ds a \int x^2 \sqrt {a x^2 + b x + c} \rd x + b \int x \sqrt {a x^2 + b x + c} \rd x + c \int \sqrt {a x^2 + b x + c} \rd x$ Linear Combination of Integrals

Thus:

 $\ds$  $\ds \int x^2 \sqrt {a x^2 + b x + c} \rd x$ $\ds$ $=$ $\ds \frac 1 {2 a} \int x \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x - \frac b {2 a} \int x \sqrt {a x^2 + b x + c} \rd x$ from $(1)$ $\ds$ $=$ $\ds \frac 1 {2 a} \paren {\frac {2 x \paren {\sqrt {a x^2 + b x + c} }^3} 3 - \frac 2 3 \int \paren {\sqrt {a x^2 + b x + c} }^3 \rd x}$ from $(3)$ $\ds$  $\, \ds - \,$ $\ds \frac b {2 a} \paren {\frac {\paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {2 a} \int \sqrt {a x^2 + b x + c} \rd x}$ Lemma for Primitive of $x \sqrt {a x^2 + b x + c}$ $\ds$ $=$ $\ds \frac {x \paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac 1 {3 a} \int \paren {\sqrt {a x^2 + b x + c} }^3 \rd x$ simplifying $\ds$  $\, \ds - \,$ $\ds \frac b {6 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {b^2} {4 a^2} \int \sqrt {a x^2 + b x + c} \rd x$ $\ds$ $=$ $\ds \frac {x \paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {6 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {b^2} {4 a^2} \int \sqrt {a x^2 + b x + c} \rd x$ $\ds$  $\, \ds - \,$ $\ds \frac 1 {3 a} \paren {a \int x^2 \sqrt {a x^2 + b x + c} \rd x + b \int x \sqrt {a x^2 + b x + c} \rd x + c \int \sqrt {a x^2 + b x + c} \rd x}$ from $(4)$ $\ds$ $=$ $\ds \frac {x \paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {6 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {b^2} {4 a^2} \int \sqrt {a x^2 + b x + c} \rd x$ simplifying $\ds$  $\, \ds - \,$ $\ds \frac 1 3 \int x^2 \sqrt {a x^2 + b x + c} \rd x - \frac b {3 a} \int x \sqrt {a x^2 + b x + c} \rd x - \frac c {3 a} \int \sqrt {a x^2 + b x + c} \rd x$ $\ds$ $=$ $\ds \frac {x \paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {6 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {b^2} {4 a^2} \int \sqrt {a x^2 + b x + c} \rd x$ $\ds$  $\, \ds - \,$ $\ds \frac 1 3 \int x^2 \sqrt {a x^2 + b x + c} \rd x - \frac c {3 a} \int \sqrt {a x^2 + b x + c} \ rd x$ $\ds$  $\, \ds - \,$ $\ds \frac b {3 a} \paren {\frac {\paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {2 a} \int \sqrt {a x^2 + b x + c} \rd x}$ Lemma for Primitive of $x \sqrt {a x^2 + b x + c}$ $\ds$ $=$ $\ds \frac {x \paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {6 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {b^2} {4 a^2} \int \sqrt {a x^2 + b x + c} \rd x$ simplifying $\ds$  $\, \ds - \,$ $\ds \frac 1 3 \int x^2 \sqrt {a x^2 + b x + c} \rd x - \frac c {3 a} \int \sqrt {a x^2 + b x + c} \rd x$ $\ds$  $\, \ds - \,$ $\ds \frac {b \paren {\sqrt {a x^2 + b x + c} }^3} {9 a^2} + \frac {b^2} {6 a^2} \int \sqrt {a x^2 + b x + c} \rd x$ $\ds$ $=$ $\ds \paren {\frac x {3 a} - \frac b {6 a^2} - \frac b {9 a^2} } \paren {\sqrt {a x^2 + b x + c} }^3$ gathering terms $\ds$  $\, \ds + \,$ $\ds \paren {\frac {b^2} {4 a^2} - \frac c {3 a} + \frac {b^2} {6 a^2} } \int \sqrt {a x^2 + b x + c} \rd x$ $\ds$  $\, \ds - \,$ $\ds \frac 1 3 \int x^2 \sqrt {a x^2 + b x + c} \rd x$

Hence:

 $\ds$  $\ds \int x^2 \sqrt {a x^2 + b x + c} \rd x + \frac 1 3 \int x^2 \sqrt {a x^2 + b x + c} \rd x$ $\ds$ $=$ $\ds \paren {\frac {6 a x} {18 a^2} - \frac {3 b} {18 a^2} - \frac {2 b} {18 a^2} } \paren {\sqrt {a x^2 + b x + c} }^3$ $\ds$  $\, \ds + \,$ $\ds \paren {\frac {3 b^2} {12 a^2} - \frac {4 a c} {12 a^2} + \frac {2 b^2} {12 a^2} } \int \sqrt {a x^2 + b x + c} \rd x$ $\ds \leadsto \ \$ $\ds \frac 4 3 \int x^2 \sqrt {a x^2 + b x + c} \rd x$ $=$ $\ds \frac {6 a x - 5 b} {18 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {5 b^2 - 4 a c} {12 a^2} \int \sqrt {a x^2 + b x + c} \rd x$ $\ds \leadsto \ \$ $\ds \int x^2 \sqrt {a x^2 + b x + c} \rd x$ $=$ $\ds \frac {6 a x - 5 b} {24 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {5 b^2 - 4 a c} {16 a^2} \int \sqrt {a x^2 + b x + c} \rd x$ multiplying by $\dfrac 3 4$

$\blacksquare$