hcju/proofwikiqa huggingface dataset:
Let: {{begin-eqn}} {{eqn | l = z       | r = x^2 }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \leadsto       | l = \int \frac {\paren {\sqrt {x^2 - a^2} }^3} {x^2} \rd x       | r = \int \frac {\paren {\sqrt {z - a^2} }^3} {2 z \sqrt z} \rd z       | c = Integration by Substitution }} {{eqn | r = \frac {-\paren {\sqrt {z - a^2} }^3} {\sqrt z} + \frac 3 2 \int \frac {\sqrt {z - a^2} } {\sqrt z} \rd z       | c = Primitive of $\dfrac {\paren {a x + b}^m} {\paren {p x + q}^n}$: Formulation 3 }} {{eqn | r = \frac {-\paren {\sqrt {x^2 - a^2} }^3} x + 3 \int \sqrt {x^2 - a^2} \rd x       | c = substituting for $z$ and simplifying }} {{eqn | r = \frac {-\paren {\sqrt {x^2 - a^2} }^3} x - \frac {3 x \sqrt {x^2 - a^2} } 2 - \frac {3 a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C       | c = Primitive of $\sqrt {x^2 - a^2}$ }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = z       | r = x^2 }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \leadsto       | l = \int \frac {\paren {\sqrt {x^2 - a^2} }^3} {x^3} \rd x       | r = \int \frac {\paren {\sqrt {z - a^2} }^3} {2 z^2} \rd z       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \paren {\frac {-\paren {\sqrt {z - a^2} }^3} z + \frac 3 2 \int \frac {\sqrt {z - a^2} } z \rd z}       | c = Primitive of $\dfrac {\paren {a x + b}^m} {\paren {p x + q}^n}$: Formulation 3 }} {{eqn | r = \frac {-\paren {\sqrt {x^2 - a^2} }^3} {2 x^2} + \frac 3 2 \int \frac {\sqrt {x^2 - a^2} } x \rd x       | c = substituting for $z$ and simplifying }} {{eqn | r = \frac {-\paren {\sqrt {x^2 - a^2} }^3} {2 x^2} + \frac 3 2 \paren {\sqrt {x^2 - a^2} - a \arcsec \size {\frac x a} } + C       | c = Primitive of $\dfrac {\sqrt {x^2 - a^2} } x$ }} {{eqn | r = \frac {-\paren {\sqrt {x^2 - a^2} }^3} {2 x^2} + \frac {3 \sqrt {x^2 - a^2} } 2 - \frac {3 a} 2 \arcsec \size {\frac x a} + C       | c = simplifying }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = z^2       | r = a^2 - x^2       | c =  }} {{eqn | ll= \leadsto       | l = 2 z \frac {\d z} {\d x}       | r = -2 x       | c = Chain Rule for Derivatives, Power Rule for Derivatives }} {{eqn | ll= \leadsto       | l = \int \frac {x \rd x} {\sqrt {a^2 - x^2} }       | r = \int -\frac {z \rd z} z       | c = Integration by Substitution }} {{eqn | r = -\int \rd z       | c =  }} {{eqn | r = -z + C       | c = Primitive of Constant }} {{eqn | r = -\sqrt {a^2 - x^2} + C       | c = substituting for $z$ }} {{end-eqn}} {{qed}}
With a view to expressing the problem in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = x^2       | c =  }} {{eqn | ll= \implies       | l = \frac {\d u} {\d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \frac x {\sqrt {a^2 - x^2} }       | c =  }} {{eqn | ll= \implies       | l = v       | r = -\sqrt {a^2 - x^2}       | c = Primitive of $\dfrac x {\sqrt {a^2 - x^2} }$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \frac {x^3 \rd x} {\sqrt {a^2 - x^2} }       | r = \int x^2 \frac {x \rd x} {\sqrt {a^2 - x^2} }       | c =  }} {{eqn | r = -x^2 \sqrt {a^2 - x^2} - \int 2 x \left({-\sqrt {a^2 - x^2} }\right) \rd x       | c = Integration by Parts }} {{eqn | r = -x^2 \sqrt {a^2 - x^2} + 2 \int x \left({\sqrt {a^2 - x^2} }\right) \rd x       | c = simplification }} {{eqn | r = -x^2 \sqrt {a^2 - x^2} + 2 \left({-\frac {\left({\sqrt {a^2 - x^2} }\right)^3} 3 }\right) + C       | c = Primitive of $x \sqrt {a^2 - x^2}$ }} {{eqn | r = \left({a^2 - x^2 - a^2}\right) \sqrt {a^2 - x^2} - \frac {2 \left({\sqrt {a^2 - x^2} }\right)^3} 3 + C       | c =  }} {{eqn | r = \left({\sqrt {a^2 - x^2} }\right)^3 - a^2 \sqrt {a^2 - x^2} - \frac {2 \left({\sqrt {a^2 - x^2} }\right)^3} 3 + C       | c =  }} {{eqn | r = \frac {\left({\sqrt {a^2 - x^2} }\right)^3} 3 - a^2 \sqrt {a^2 - x^2} + C       | c =  }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = z       | r = x^2 }} {{eqn | ll= \implies       | l = \frac {\mathrm d z} {\mathrm d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \implies       | l = \int \frac {\mathrm d x} {x^2 \sqrt {a^2 - x^2} }       | r = \int \frac {\mathrm d z} {2 z \sqrt z \sqrt {a^2 - z} }       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \frac {\mathrm d z} {z^{3/2} \sqrt {a^2 - z} }       | c =  }} {{end-eqn}} Using Primitive of $ \dfrac 1{x^m \sqrt{a x + b} }$: :$\displaystyle \int \frac {\mathrm d x} {x^m \sqrt{a x + b} } = -\frac {\sqrt{a x + b} } {\left({m - 1}\right) b x^{m-1} } - \frac {\left({2 m - 3}\right) a} {\left({2 m - 2}\right) b} \int \frac {\mathrm d x} {x^{m - 1} \sqrt{a x + b} }$ Setting: {{begin-eqn}} {{eqn | l = x       | o = :=       | r = -z }} {{eqn | l = m       | o = :=       | r = \frac 3 2 }} {{eqn | l = a       | o = :=       | r = 1 }} {{eqn | l = b       | o = :=       | r = a^2 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \frac 1 2 \int \frac {\mathrm d z} {\left({-z}\right)^{3/2} \sqrt {-z + a^2} }       | r = \frac {-\sqrt{-z + a^2} } {2 \left({\left({\frac 3 2}\right) - 1}\right) a^2 \left({-z}\right)^{\left({3/2}\right) - 1} } - \frac {2 \left({\frac 3 2}\right) - 3} {2 \left({2 \left({\frac 3 2}\right) - 2}\right) a^2} \int \frac {\mathrm d z} {\left({-z}\right)^{\left({\frac 3 2}\right) - 1} \sqrt{-z + a^2} } + C       | c = Primitive of $ \dfrac 1{x^m \sqrt{a x + b} }$ }} {{eqn | r = \frac {-\sqrt{a^2 - z} } {a^2 z^{1/2} } - 0 + C       | c = simplifying }} {{eqn | r = \frac {-\sqrt {a^2 - x^2} } {a^2 x} + C       | c = substituting back for $z$ }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = z       | r = x^2 }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \leadsto       | l = \int \frac {\d x} {x^3 \sqrt {a^2 - x^2} }       | r = \int \frac {\d z} {2 z^{3/2} \sqrt z \sqrt {a^2 - z} }       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \frac {\d z} {z^2 \sqrt {-z + a^2} }       | c =  }} {{end-eqn}} Using Primitive of $ \dfrac 1{x^m \sqrt{a x + b} }$: :$\displaystyle \int \frac {\d x} {x^m \sqrt {a x + b} } = -\frac {\sqrt{a x + b} } {\paren {m - 1} b x^{m - 1} } - \frac {\paren {2 m - 3} a} {\paren {2 m - 2} b} \int \frac {\d x} {x^{m - 1} \sqrt{a x + b} }$ Setting: {{begin-eqn}} {{eqn | l = x       | o = :=       | r = z }} {{eqn | l = m       | o = :=       | r = 2 }} {{eqn | l = a       | o = :=       | r = -1 }} {{eqn | l = b       | o = :=       | r = a^2 }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \frac 1 2 \int \frac {\d z} {z^2 \sqrt {-z + a^2} }       | r = \frac {-\sqrt {-z + a^2} } {2 \paren {2 - 1} a^2 z^{2 - 1} } - \frac {\paren {2 \paren 2 - 3} \paren {-1} } {2 \paren {2 \paren 2 - 2} a^2} \int \frac {\d z} {z^{2 - 1} \sqrt {-z + a^2} } + C       | c = Primitive of $ \dfrac 1{x^m \sqrt{a x + b} }$ }} {{eqn | r = \frac {-\sqrt {a^2 - z} } {2 a^2 z} - \frac 1 {4 a^2} \int \frac {\d z} {z \sqrt{a^2 - z} } + C       | c = simplifying }} {{eqn | r = \frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} + \frac 1 {4 a^2} \paren {\int \frac {2 x \rd x} {x^2 \sqrt {a^2 - x^2} } } + C       | c = substituting for $z$ }} {{eqn | r = \frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} + \frac 1 {2 a^2} \paren {\int \frac {\d x} {x \sqrt {a^2 - x^2} } } + C       | c = simplification }} {{eqn | r =\frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} + \frac 1 {2 a^3} \paren {-\map \ln {\frac {a + \sqrt {a^2 - x^2} } x} } + C       | c = Primitive of $\dfrac 1 {x \sqrt {a^2 - x^2} }$ }} {{eqn | r =\frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} - \frac 1 {2 a^3} \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C       | c = simplifying }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = x       | r = a \sin \theta }} {{eqn | n = 1       | ll= \leadsto       | l = \frac {\d x} {\d \theta}       | r = a \cos \theta       | c = Derivative of Sine Function }} {{end-eqn}} Also: {{begin-eqn}} {{eqn | l = x       | r = a \sin \theta }} {{eqn | ll= \leadsto       | l = a^2 - x^2       | r = a^2 - a^2 \sin^2 \theta       | c =  }} {{eqn | r = a^2 \paren {1 - \sin^2 \theta}       | c =  }} {{eqn | r = a^2 \cos^2 \theta       | c = Sum of Squares of Sine and Cosine }} {{eqn | n = 2       | ll= \leadsto       | l = \sqrt {a^2 - x^2}       | r = a \cos \theta       | c =  }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = x       | r = a \sin \theta }} {{eqn | n = 3       | ll= \leadsto       | l = \theta       | r = \map \arcsin {\frac x a}       | c = {{Defof|Real Arcsine}} }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \int \sqrt {a^2 - x^2} \rd x       | r = \int \sqrt {a^2 - x^2} \, a \cos \theta \rd \theta       | c = Integration by Substitution from $(1)$ }} {{eqn | r = \int a^2 \cos^2 \theta \rd \theta       | c = substituting for $\sqrt {a^2 - x^2}$ from $(2)$ }} {{eqn | r = a^2 \int \cos^2 \theta \rd \theta       | c = Primitive of Constant Multiple of Function }} {{eqn | r = a^2 \frac {\sin \theta \cos \theta + \theta} 2 + C       | c = Primitive of Square of Cosine Function: Corollary }} {{eqn | r = \frac 1 2 a \sin \theta \, a \cos \theta + \frac {a^2 \theta} 2 + C       | c = rearranging }} {{eqn | r = \frac 1 2 x a \cos \theta + \frac {a^2 \theta} 2 + C       | c = substituting $x = a \sin \theta$ }} {{eqn | r = \frac 1 2 x \sqrt {a^2 - x^2} + \frac {a^2 \theta} 2 + C       | c = substituting $\sqrt {a^2 - x^2} = a \cos \theta$ from $(2)$ }} {{eqn | r = \frac {x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \map \arcsin {\frac x a} + C       | c = substituting for $\theta = \map \arcsin {\dfrac x a}$ from $(3)$ }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = z       | r = x^2 }} {{eqn | ll= \implies       | l = \frac {\mathrm d z} {\mathrm d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \implies       | l = \int x \sqrt {a^2 - x^2} \ \mathrm d x       | r = \int \frac {\sqrt z \sqrt {a^2 - z} \ \mathrm d z} {2 \sqrt z}       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \sqrt {a^2 - z} \ \mathrm d z       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \frac {2 \left({\sqrt {a^2 - z} }\right)^3} {-3} + C       | c = Primitive of $\sqrt {a x + b}$ }} {{eqn | r = \frac {-\left({\sqrt {a^2 - x^2} }\right)^3} 3 + C       | c = substituting for $z$ and simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = z       | r = x^2 }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \leadsto       | l = \int x^2 \sqrt {a^2 - x^2} \rd x       | r = \int \frac {z \sqrt {a^2 - z} \rd z} {2 \sqrt z}       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \sqrt z \sqrt {a^2 - z} \rd z       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \paren {\frac {-2 z + a^2} 4 \sqrt z \sqrt {a^2 - z} + \frac {a^4} 8 \int \frac {\d z} {\sqrt z \sqrt {a^2 - z} } } + C       | c = Primitive of $\sqrt {\paren {a x + b} \paren {p x + q} }$ }} {{eqn | r = \frac {-2 z + a^2} 8 \sqrt z \sqrt {a^2 - z} + \frac {a^4} {16} \paren {2 \sinh^{-1} \sqrt {\frac z {a^2} } } + C       | c = Primitive of $\dfrac 1 {\sqrt {\paren {a x + b} \paren {p x + q} } }$ }} {{eqn | r = \frac {-2 z + 2 a^2} 8 \sqrt z \sqrt {a^2 - z} + \frac {a^2} 8 \sqrt z \sqrt {a^2 - z} + \frac {a^4} 8 \sinh^{-1} \sqrt {\frac z {a^2} } + C       | c =  }} {{eqn | r = \frac {a^2 - z} 4 \sqrt z \sqrt {a^2 - z} + \frac {a^2} 8 \sqrt z \sqrt {a^2 - z} + \frac {a^4} 8 \sinh^{-1} \frac {\sqrt z} a + C       | c =  }} {{eqn | r = \frac {-x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {a^2 x \sqrt {x^2 - a^2} } 8 + \frac {a^4} 8 \sinh^{-1} \frac x a + C       | c = substituting for $z$ }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = z       | r = x^2 }} {{eqn | ll= \implies       | l = \frac {\mathrm d z} {\mathrm d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \implies       | l = \int x^3 \sqrt {a^2 - x^2} \ \mathrm d x       | r = \int \frac {z^{3/2} \sqrt {a^2 - z} \ \mathrm d z} {2 \sqrt z}       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int z \sqrt {a^2 - z} \ \mathrm d z       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \left({\frac {2 \left({-3 z - 2 a^2}\right)} {15} \left({\sqrt {a^2 - z} }\right)^3 }\right) + C       | c = Primitive of $x \sqrt {\left({a x + b}\right)}$ }} {{eqn | r = \frac {-3 z - 2 a^2} {15} \left({\sqrt {a^2 - z} }\right)^3 + C       | c = simplification }} {{eqn | r = \frac {-3 x^2 - 2 a^2} {15} \left({\sqrt {a^2 - x^2} }\right)^3 + C       | c = substituting for $z$ }} {{eqn | r = \frac {3 a^2 - 3 x^2 - 5 a^2} {15} \left({\sqrt {a^2 - x^2} }\right)^3 + C       | c =  }} {{eqn | r = \frac {3 \left({a^2 - x^2}\right) \left({\sqrt {a^2 - x^2} }\right)^3 - 5 a^2 \left({\sqrt {a^2 - x^2} }\right)^3} {15} + C       | c =  }} {{eqn | r = \frac {\left({\sqrt {a^2 - x^2} }\right)^5} 5 - \frac {a^2 \left({\sqrt {a^2 - x^2} }\right)^3} 3 + C       | c = simplification }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = z       | r = x^2 }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \leadsto       | l = \int \frac {\sqrt {a^2 - x^2} } x \rd x       | r = \int \frac {\sqrt {a^2 - z} \rd z} {2 \sqrt z \sqrt z}       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \frac {\sqrt {a^2 - z} \rd z} z       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \paren {2 \sqrt {a^2 - z} + a^2 \int \frac {\d z} {z \sqrt {a^2 - z} } } + C       | c = Primitive of $\dfrac {\sqrt {a x + b} } x$ }} {{eqn | r = \sqrt {a^2 - x^2} + \frac {a^2} 2 \int \frac {2 x \rd x} {x^2 \sqrt {a^2 - x^2} } + C       | c = substituting for $z$ }} {{eqn | r = \sqrt {a^2 - x^2} + a^2 \int \frac {\d x} {x \sqrt {a^2 - x^2} } + C       | c = simplifying }} {{eqn | r = \sqrt {a^2 - x^2} - a^2 \paren {-\frac 1 a \, \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} } + C       | c = Primitive of $\dfrac 1 {x \sqrt {a^2 - x^2} }$ }} {{eqn | r = \sqrt {a^2 - x^2} - a \, \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C       | c = simplification }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = z       | r = x^2 }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \leadsto       | l = \int \frac {\sqrt {a^2 - x^2} } {x^2} \rd x       | r = \int \frac {\sqrt {a^2 - z} \rd z} {2 z \sqrt z}       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \frac {\sqrt {a^2 - z} \rd z} {z^{3/2} }       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \paren {\frac {-\sqrt {a^2 - z} } {\frac 1 2 \sqrt z} - \frac 1 2 \int \frac {\d z} {\sqrt z \sqrt {a^2 - z} } } + C       | c = Primitive of $\dfrac {\sqrt {a x + b} } {x^m}$ }} {{eqn | r = \frac {-\sqrt {a^2 - x^2} } x - \frac 1 2 \int \frac {2 x \rd x} {x \sqrt {a^2 - x^2} } + C       | c = substituting for $z$ }} {{eqn | r = \frac {-\sqrt {a^2 - x^2} } x - \int \frac {\d x} {\sqrt {a^2 - x^2} } + C       | c = simplifying }} {{eqn | r = \frac {-\sqrt {a^2 - x^2} } x - \arcsin \frac x a + C       | c = Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$ }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = z       | r = x^2 }} {{eqn | ll= \implies       | l = \frac {\mathrm d z} {\mathrm d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \implies       | l = \int \frac {\mathrm d x} {\left({\sqrt {a^2 - x^2} }\right)^3}       | r = \int \frac {\mathrm d z} {2 \sqrt z \left({\sqrt {a^2 - z} }\right)^3}       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \frac {\mathrm d z} {\left({a^2 - z}\right) \sqrt z \sqrt {a^2 - z} }       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \left({\frac {2 \sqrt z} {a^2 \sqrt {a^2 - z} } }\right) + C       | c = Primitive of $\dfrac 1 {\left({p x + q}\right) \sqrt{\left({a x + b}\right) \left({p x + q}\right)} }$ }} {{eqn | r = \frac 1 2 \left({\frac {2 x} {a^2 \sqrt {a^2 - x^2} } }\right) + C       | c = substituting for $z$ }} {{eqn | r = \frac x {a^2 \sqrt {a^2 - x^2} } + C       | c = simplifying }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = z       | r = a^2 - x^2 }} {{eqn | ll= \implies       | l = \frac {\mathrm d z} {\mathrm d x}       | r = -2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \implies       | l = \int \frac {x \ \mathrm d x} {\left({\sqrt {a^2 - x^2} }\right)^3}       | r = \int \frac {x \mathrm d z} {-2 x z^{3/2} }       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int -z^{-3/2} \ \mathrm d z       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \left({\frac {z^{1/2} } {\frac 1 2}  }\right) + C       | c = Primitive of Power }} {{eqn | r = \frac 1 {\sqrt z} + C       | c = simplifying }} {{eqn | r = \frac 1 {\sqrt {a^2 - x^2} } + C       | c = substituting for $z$ }} {{end-eqn}} {{qed}}
With a view to expressing the problem in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = x }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = 1       | c = Power Rule for Derivatives }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \frac x {\paren {\sqrt {a^2 - x^2} }^3} }} {{eqn | ll= \leadsto       | l = v       | r = \frac 1 {\sqrt {a^2 - x^2} }       | c = Primitive of $\dfrac x {\paren {\sqrt {a^2 - x^2} }^3}$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \frac {x^2 \rd x} {\paren {\sqrt {a^2 - x^2} }^3}       | r = \int x \frac {x \rd x} {\paren {\sqrt {a^2 - x^2} }^3}       | c =  }} {{eqn | r = x \frac 1 {\sqrt {a^2 - x^2} } - \int \frac 1 {\sqrt {a^2 - x^2} } 1 + C       | c = Integration by Parts }} {{eqn | r = \frac x {\sqrt {a^2 - x^2} } - \int \frac 1 {\sqrt {a^2 - x^2} } + C       | c = simplifying }} {{eqn | r = \frac x {\sqrt {a^2 - x^2} } - \arcsin \frac x a + C       | c = Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$ }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = z       | r = x^2 }} {{eqn | ll= \implies       | l = \frac {\mathrm d z} {\mathrm d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \implies       | l = \int \frac {\mathrm d x} {x \left({\sqrt {a^2 - x^2} }\right)^3}       | r = \int \frac {\mathrm d z} {2 \sqrt z \sqrt z \left({\sqrt {a^2 - z} }\right)^3}       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \frac {\mathrm d z} {z \left({\sqrt {a^2 - z} }\right)^3}       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \left({\frac 2 {a^2 \sqrt {a^2 - z} } + \frac 1 {a^2} \int \frac {\mathrm d z} {z \sqrt {a^2 - z} } }\right) + C       | c = Primitive of $\dfrac 1 {x \left({\sqrt{a x + b} }\right)^m }$ }} {{eqn | r = \frac 1 {a^2 \sqrt {a^2 - z} } + \frac 1 {2 a^2} \int \frac {\mathrm d z} {z \sqrt {a^2 - z} } + C       | c = simplifying }} {{eqn | r = \frac 1 {a^2 \sqrt {a^2 - x^2} } + \frac 1 {2 a^2} \int \frac {2 x \mathrm d x} {x^2 \sqrt {a^2 - x^2} } + C       | c = substituting for $z$ }} {{eqn | r = \frac 1 {a^2 \sqrt {a^2 - x^2} } + \frac 1 {a^2} \int \frac {\mathrm d x} {x \sqrt {a^2 - x^2} } + C       | c = simplifying }} {{eqn | r = \frac1 {a^2 \sqrt {a^2 - x^2} } - \frac 1 {a^3} \ln \left({\frac {a + \sqrt {a^2 - x^2} } x}\right) + C       | c = Primitive of $\dfrac 1 {x \sqrt {a^2 - x^2} }$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {x^2 \left({\sqrt {a^2 - x^2} }\right)^3}       | r = \int \frac {a^2 \ \mathrm d x} {a^2 x^2 \left({\sqrt {a^2 - x^2} }\right)^3}       | c =  }} {{eqn | r = \int \frac {\left({a^2 - x^2 + x^2}\right) \ \mathrm d x} {a^2 x^2 \left({\sqrt {a^2 - x^2} }\right)^3}       | c =  }} {{eqn | r = \frac 1 {a^2} \int \frac {\left({a^2 - x^2}\right) \ \mathrm d x} {a^2 x^2 \left({\sqrt {a^2 - x^2} }\right)^3} + \frac 1 {a^2} \int \frac {x^2 \ \mathrm d x} {x^2 \left({\sqrt {a^2 - x^2} }\right)^3}       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 {a^2} \int \frac {\mathrm d x} {x^2 \sqrt {a^2 - x^2} } + \frac 1 {a^2} \int \frac {\mathrm d x} {\left({\sqrt {a^2 - x^2} }\right)^3}       | c = simplifying }} {{eqn | r = \frac 1 {a^2} \frac {-\sqrt {a^2 - x^2} } {a^2 x} + \frac 1 {a^2} \int \frac {\mathrm d x} {\left({\sqrt {x^2 - a^2} }\right)^3} + C       | c = Primitive of $\dfrac 1 {x^2 \sqrt {a^2 - x^2} }$ }} {{eqn | r = \frac {-\sqrt {a^2 - x^2} } {a^4 x} + \frac 1 {a^2} \frac x {a^2 \sqrt {a^2 - x^2} } + C       | c = Primitive of $\dfrac 1 {\left({\sqrt {a^2 - x^2} }\right)^3}$ }} {{eqn | r = \frac {-\sqrt {a^2 - x^2} } {a^4 x} + \frac x {a^4 \sqrt {x^2 - a^2} } + C       | c = simplification }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {x^3 \left({\sqrt {a^2 - x^2} }\right)^3}       | r = \int \frac {a^2 \ \mathrm d x} {a^2 x^3 \left({\sqrt {a^2 - x^2} }\right)^3}       | c =  }} {{eqn | r = \int \frac {\left({a^2 - x^2 + x^2}\right) \ \mathrm d x} {a^2 x^3 \left({\sqrt {a^2 - x^2} }\right)^3}       | c =  }} {{eqn | r = \frac 1 {a^2} \int \frac {\left({a^2 - x^2}\right) \ \mathrm d x} {x^3 \left({\sqrt {a^2 - x^2} }\right)^3} + \frac 1 {a^2} \int \frac {x^2 \ \mathrm d x} {x^3 \left({\sqrt {a^2 - x^2} }\right)^3}       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 {a^2} \int \frac {\mathrm d x} {x^3 \sqrt {a^2 - x^2} } + \frac 1 {a^2} \int \frac {\mathrm d x} {x \left({\sqrt {a^2 - x^2} }\right)^3}       | c = simplifying }} {{eqn | r = \frac 1 {a^2} \left({\frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} - \frac 1 {2 a^3} \ln \left({\frac {a + \sqrt {a^2 - x^2} } x}\right)}\right) + \frac 1 {a^2} \int \frac {\mathrm d x} {x \left({\sqrt {x^2 - a^2} }\right)^3} + C       | c = Primitive of $\dfrac 1 {x^3 \sqrt {a^2 - x^2} }$ }} {{eqn | r = \frac 1 {a^2} \left({\frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} - \frac 1 {2 a^3} \ln \left({\frac {a + \sqrt {a^2 - x^2} } x}\right)}\right)       | c =  }} {{eqn | o =        | ro= +       | r = \frac 1 {a^2} \left({\frac 1 {a^2 \sqrt {a^2 - x^2} } - \frac 1 {a^3} \ln \left({\frac {a + \sqrt {a^2 - x^2} } x}\right)}\right) + C       | c = Primitive of $\dfrac 1 {x \left({\sqrt {a^2 - x^2} }\right)^3}$ }} {{eqn | r = \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} \ln \left({\frac {a + \sqrt {a^2 - x^2} } x}\right) + C       | c = simplification }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = z       | r = x^2 }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \leadsto       | l = \int \paren {\sqrt {a^2 - x^2} }^3 \rd x       | r = \int \frac {\paren {\sqrt {a^2 - z} }^3} {2 \sqrt z} \rd x       | c = Integration by Substitution }} {{eqn | r = \frac {\sqrt z \paren {\sqrt {a^2 - z} }^3} 4 + \frac {3 a^2} 8 \int \frac {\sqrt {a^2 - z} } {\sqrt z} \rd x + C       | c = Primitive of $\dfrac {\paren {p x + q}^n} {\sqrt{a x + b} }$ }} {{eqn | r = \frac {\sqrt z \paren {\sqrt {a^2 - z} }^3} 4 + \frac {3 a^2} 8 \paren {\sqrt z \sqrt {a^2 - z} + \frac {a^2} 2 \int \frac {\d x} {\sqrt z \sqrt {a^2 - z} } } + C       | c = Primitive of $\dfrac {\sqrt{p x + q} } {\sqrt{a x + b} }$ }} {{eqn | r = \frac {x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {3 a^2} 8 x \sqrt {a^2 - x^2} + \frac {3 a^4} 8 \int \frac {\d x} {\sqrt {a^2 - x^2} } + C       | c = substituting for $z$ and simplifying }} {{eqn | r = \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       | c = Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$ }} {{end-eqn}} {{qed}}