Primitive of Reciprocal of Root of x squared plus a squared cubed
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Theorem
- $\ds \int \frac {\d x} {\paren {\sqrt {x^2 + a^2} }^3} = \frac x {a^2 \sqrt {x^2 + a^2} } + C$
Proof 1
\(\ds x\) | \(=\) | \(\ds a \tan \theta\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d x} {\d \theta}\) | \(=\) | \(\ds a \sec^2 \theta\) | Derivative of Tangent Function | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {\paren {\sqrt {x^2 + a^2} }^3}\) | \(=\) | \(\ds \int \frac {a \sec^2 \theta \rd \theta} {\sqrt {a^2 \tan^2 \theta + a^2}^3}\) | Integration by Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {a \sec^2 \theta \rd \theta} {a^3 \sec^3 \theta}\) | Difference of Squares of Secant and Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {a^2} \int \cos \theta \rd \theta\) | Definition of Real Secant Function and simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {a^2} \sin \theta + C\) | Primitive of $\cos x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {a^2} \dfrac {a \sin \theta} {\cos \theta} \dfrac {\cos \theta} a + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {a^2} \dfrac {a \tan \theta} {a \sec \theta} + C\) | Tangent is Sine divided by Cosine, Definition of Real Secant Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {a^2} \dfrac {a \tan \theta} {\sqrt {a^2 \sec^2 \theta} } + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {a^2} \dfrac {a \tan \theta} {\sqrt {a^2 \tan^2 \theta + a^2} } + C\) | Difference of Squares of Secant and Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {a^2} \dfrac x {\sqrt {x^2 + a^2} } + C\) | substituting $x = a \tan \theta$ |
$\blacksquare$
Proof 2
\(\ds z\) | \(=\) | \(\ds x^2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds 2 x\) | Power Rule for Derivatives | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {\paren {\sqrt {x^2 + a^2} }^3}\) | \(=\) | \(\ds \int \frac {\d z} {2 \sqrt z \paren {\sqrt {z + a^2} }^3}\) | Integration by Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int \frac {\d z} {\paren {z + a^2} \sqrt z \sqrt {z + a^2} }\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\frac {2 \sqrt z} {a^2 \sqrt {z + a^2} } } + C\) | Primitive of $\dfrac 1 {\paren {p x + q} \sqrt {\paren {a x + b} \paren {p x + q} } }$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\frac {2 x} {a^2 \sqrt {x^2 + a^2} } } + C\) | substituting for $z$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac x {a^2 \sqrt {x^2 + a^2} } + C\) | simplifying |
$\blacksquare$
Also see
- Primitive of $\dfrac 1 {\paren {\sqrt {x^2 - a^2} }^3}$
- Primitive of $\dfrac 1 {\paren {\sqrt {a^2 - x^2} }^3}$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.196$