Primitive of x cubed by Root of a squared minus x squared cubed
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Theorem
- $\ds \int x^3 \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {\paren {\sqrt {a^2 - x^2} }^7} 7 - \frac {a^2 \paren {\sqrt {a^2 - x^2} }^5} 5 + C$
Proof
\(\ds \int x^3 \paren {\sqrt {a^2 - x^2} }^3 \rd x\) | \(=\) | \(\ds \int x \paren {x^2} \paren {\sqrt {a^2 - x^2} }^3 \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int x \paren {x^2 - a^2 + a^2} \paren {\sqrt {a^2 - x^2} }^3 \rd x\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds -\int x \paren {a^2 - x^2} \paren {\sqrt {a^2 - x^2} }^3 \rd x + a^2 \int x \paren {\sqrt {a^2 - x^2} }^3 \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds -\int x \paren {\sqrt {a^2 - x^2} }^5 \rd x + a^2 \int x \paren {\sqrt {a^2 - x^2} }^3 \rd x\) | simplifying |
Let:
\(\ds z\) | \(=\) | \(\ds a^2 - x^2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds -2 x\) | Power Rule for Derivatives | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds -\int x \paren {\sqrt {a^2 - x^2} }^5 \rd x + a^2 \int x \paren {\sqrt {a^2 - x^2} }^3 \rd x\) | |||||||||||
\(\ds \) | \(=\) | \(\ds -\int \frac {\sqrt z z^{5/2} } {-2 \sqrt z} \rd z + a^2 \int \frac {\sqrt z z^{3/2} } {-2 \sqrt z} \rd z\) | Integration by Substitution | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int z^{5/2} \rd z - \frac {a^2} 2 \int z^{3/2} \rd z\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \frac {z^{7/2} } {7/2} - \frac {a^2} 2 \frac {z^{5/2} } {5/2} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {z^{7/2} } 7 - a^2 \frac {z^{5/2} } 5 + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {\sqrt {a^2 - x^2} }^7} 7 - \frac {a^2 \paren {\sqrt {a^2 - x^2} }^5} 5 + C\) | substituting for $z$ |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {a^2 - x^2}$: $14.261$