Primitives involving Root of a squared minus x squared cubed
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Theorem
This page gathers together the primitives of some expressions involving $\left({\sqrt{a^2 - x^2} }\right)^3$.
Primitive of Reciprocal of $\left({\sqrt{a^2 - x^2} }\right)^3$
- $\ds \int \frac {\d x} {\paren {\sqrt {a^2 - x^2} }^3} = \frac x {a^2 \sqrt {a^2 - x^2} } + C$
Primitive of $x$ over $\left({\sqrt{a^2 - x^2} }\right)^3$
- $\ds \int \frac {x \rd x} {\paren {\sqrt {a^2 - x^2} }^3} = \frac 1 {\sqrt {a^2 - x^2} } + C$
Primitive of $x^2$ over $\left({\sqrt{a^2 - x^2} }\right)^3$
- $\ds \int \frac {x^2 \rd x} {\paren {\sqrt {a^2 - x^2} }^3} = \frac x {\sqrt {a^2 - x^2} } - \arcsin \frac x a + C$
Primitive of $x^3$ over $\left({\sqrt{a^2 - x^2} }\right)^3$
- $\ds \int \frac {x^3 \rd x} {\paren {\sqrt {a^2 - x^2} }^3} = \sqrt {a^2 - x^2} + \frac {a^2} {\sqrt {a^2 - x^2} } + C$
Primitive of Reciprocal of $x \left({\sqrt{a^2 - x^2} }\right)^3$
- $\ds \int \frac {\d x} {x \paren {\sqrt {a^2 - x^2} }^3} = \frac 1 {a^2 \sqrt {a^2 - x^2} } - \frac 1 {a^3} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$
Primitive of Reciprocal of $x^2 \left({\sqrt{a^2 - x^2} }\right)^3$
- $\ds \int \frac {\d x} {x^2 \paren {\sqrt {a^2 - x^2} }^3} = \frac {-\sqrt {a^2 - x^2} } {a^4 x} + \frac x {a^4 \sqrt {a^2 - x^2} } + C$
Primitive of Reciprocal of $x^3 \left({\sqrt{a^2 - x^2} }\right)^3$
- $\ds \int \frac {\d x} {x^3 \paren {\sqrt {a^2 - x^2} }^3} = \frac {-1} {2 a^2 x^2 \sqrt {a^2 - x^2} } + \frac 3 {2 a^4 \sqrt {a^2 - x^2} } - \frac 3 {2 a^5} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$
Primitive of $\left({\sqrt{a^2 - x^2} }\right)^3$
- $\ds \int \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {3 a^2 x \sqrt {a^2 - x^2} } 8 + \frac {3 a^4} 8 \arcsin \frac x a + C$
Primitive of $x \left({\sqrt{a^2 - x^2} }\right)^3$
- $\ds \int x \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {-\paren {\sqrt {a^2 - x^2} }^5} 5 + C$
Primitive of $x^2 \left({\sqrt{a^2 - x^2} }\right)^3$
- $\ds \int x^2 \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {-x \paren {\sqrt {a^2 - x^2} }^5} 6 + \frac {a^2 x \paren {\sqrt {a^2 - x^2} }^3} {24} + \frac {a^4 x \sqrt {a^2 - x^2} } {16} + \frac {a^6} {16} \arcsin \frac x a + C$
Primitive of $x^3 \left({\sqrt{a^2 - x^2} }\right)^3$
- $\ds \int x^3 \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {\paren {\sqrt {a^2 - x^2} }^7} 7 - \frac {a^2 \paren {\sqrt {a^2 - x^2} }^5} 5 + C$
Primitive of $\left({\sqrt{a^2 - x^2} }\right)^3$ over $x$
- $\ds \int \frac {\paren {\sqrt {a^2 - x^2} }^3} x \rd x = \frac {\paren {\sqrt {a^2 - x^2} }^3} 3 + a^2 \sqrt {a^2 - x^2} - a^3 \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$
Primitive of $\left({\sqrt{a^2 - x^2} }\right)^3$ over $x^2$
- $\ds \int \frac {\paren {\sqrt {a^2 - x^2} }^3} {x^2} \rd x = \frac {-\paren {\sqrt {a^2 - x^2} }^3} x - \frac {3 x \sqrt {a^2 - x^2} } 2 - \frac {3 a^2} 2 \arcsin \frac x a + C$
Primitive of $\left({\sqrt{a^2 - x^2} }\right)^3$ over $x^3$
- $\ds \int \frac {\paren {\sqrt {a^2 - x^2} }^3} {x^3} \rd x = \frac {-\paren {\sqrt {a^2 - x^2} }^3} {2 x^2} - \frac {3 \sqrt {a^2 - x^2} } 2 + \frac {3 a} 2 \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$