Primitive of Root of x squared plus a squared cubed over x cubed

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Theorem

$\displaystyle \int \frac{\left({\sqrt {x^2 + a^2} }\right)^3} {x^3} \ \mathrm d x = \frac {-\left({\sqrt {x^2 + a^2} }\right)^3} {2 x^2} + \frac {3 \sqrt {x^2 + a^2} } 2 - \frac {3 a} 2 \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) + C$


Proof

Let:

\(\displaystyle z\) \(=\) \(\displaystyle x^2\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d z} {\mathrm d x}\) \(=\) \(\displaystyle 2 x\) Power Rule for Derivatives
\(\displaystyle \implies \ \ \) \(\displaystyle \int \frac{\left({\sqrt {x^2 + a^2} }\right)^3} {x^3} \ \mathrm d x\) \(=\) \(\displaystyle \int \frac{\left({\sqrt {z + a^2} }\right)^3} {2 z^2} \ \mathrm d z\) Integration by Substitution
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \left({\frac {-\left({\sqrt {z + a^2} }\right)^3} z + \frac 3 2 \int \frac{\sqrt {z + a^2} } z \ \mathrm d z}\right)\) Primitive of $\dfrac {\left({a x + b}\right)^m} {\left({p x + q}\right)^n}$: Formulation 3
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\left({\sqrt {x^2 + a^2} }\right)^3} {2 x^2} + \frac 3 2 \int \frac {\sqrt {x^2 + a^2} } x \ \mathrm d x\) substituting for $z$ and simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\left({\sqrt {x^2 + a^2} }\right)^3} {2 x^2} + \frac 3 2 \left({\sqrt {x^2 + a^2} - a \ln \left({\frac {a + \sqrt {x^2 + a^2} } a}\right)}\right) + C\) Primitive of $\dfrac {\sqrt {x^2 + a^2} } x$
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\left({\sqrt {x^2 + a^2} }\right)^3} {2 x^2} + \frac {3 \sqrt {x^2 + a^2} } 2 - \frac {3 a} 2 \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) + C\) simplifying

$\blacksquare$


Also see


Sources