Primitive of x squared by Root of a x squared plus b x plus c
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Theorem
Let $a \in \R_{\ne 0}$.
Then:
- $\ds \int x^2 \sqrt {a x^2 + b x + c} \rd x = \frac {6 a x - 5 b} {24 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {5 b^2 - 4 a c} {16 a^2} \int \sqrt {a x^2 + b x + c} \rd x$
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 Primitives |
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 $\ds \int x \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x$ in the form:
- $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
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 Primitives |
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 $\dfrac 3 4$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {a x^2 + bx + c}$: $14.287$