hcju/proofwikiqa huggingface dataset:
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= \leadsto       | 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 {x^2 - a^2} }       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \sqrt {x^2 - a^2}       | c = Primitive of $\dfrac x {\sqrt {x^2 - a^2} }$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \frac {x^3 \rd x} {\sqrt {x^2 - a^2} }       | r = \int x^2 \frac {x \rd x} {\sqrt {x^2 - a^2} }       | c =  }} {{eqn | r = x^2 \sqrt {x^2 - a^2} - \int 2 x \sqrt {x^2 - a^2} \rd x       | c = Integration by Parts }} {{eqn | r = x^2 \sqrt {x^2 - a^2} - 2 \paren {\frac {\paren {\sqrt {x^2 - a^2} }^3} 3} + C       | c = Primitive of $x \sqrt {x^2 - a^2}$ }} {{eqn | r = \paren {x^2 - a^2 + a^2} \sqrt {x^2 - a^2} - 2 \paren {\frac {\paren {\sqrt {x^2 - a^2} }^3} 3} + C       | c =  }} {{eqn | r = \paren {\sqrt {x^2 - a^2} }^3 + a^2 \sqrt {x^2 - a^2} - 2 \paren {\frac {\paren {\sqrt {x^2 - a^2} }^3} 3} + C       | c =  }} {{eqn | r = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 + a^2 \sqrt {x^2 - a^2} + C       | c =  }} {{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^2 \sqrt {x^2 - a^2} }       | r = \int \frac {\d z} {2 z \sqrt z \sqrt {z - a^2} }       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \frac {\d z} {z^{3/2} \sqrt {z - a^2} }       | c =  }} {{end-eqn}} Using Primitive of $\dfrac 1 {u^m \sqrt {a u + b} }$: :$\displaystyle \int \frac {\d u} {u^m \sqrt {a u + b} } = -\frac {\sqrt {a u + b} } {\paren {m - 1} b u^{m - 1} } - \frac {\paren {2 m - 3} a} {\paren {2 m - 2} b} \int \frac {\d u} {u^{m - 1} \sqrt {a u + b} }$ Setting: {{begin-eqn}} {{eqn | l = u       | 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 {\d z} {z^{3/2} \sqrt {z - a^2} }       | r = \frac {-\sqrt {z - a^2} } {2 \paren {\paren {\frac 3 2} - 1} \paren {-a^2} z^{\paren {3/2} - 1} } - \frac {2 \paren {\frac 3 2} - 3} {2 \paren {2 \paren {\frac 3 2} - 2} \paren {-a^2} } \int \frac {\d z} {z^{\paren {\frac 3 2} - 1} \sqrt{z - a^2} } + C       | c = Primitive of $ \dfrac 1 {x^m \sqrt {a x + b} }$ }} {{eqn | r = \frac {-\sqrt {z - a^2} } {-a^2 z^{1/2} } - 0 + C       | c = simplifying }} {{eqn | r = \frac {\sqrt {x^2 - a^2} } {a^2 x} + C       | c = substituting $x$ 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 {x^2 - a^2} }       | r = \int \frac {\d z} {2 z^{3/2} \sqrt z \sqrt {z - a^2} }       | 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} \paren {-a^2} z^{2 - 1} } - \frac {2 \paren 2 - 3} {2 \paren {2 \paren 2 - 2} \paren {-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 {z - a^2} } {2 a^2 z} + \frac 1 {4 a^2} \int \frac {\d z} {z \sqrt{z - a^2} } + C       | c = simplifying }} {{eqn | r = \frac {\sqrt {x^2 - a^2} } {2 a^2 x^2} + \frac 1 {4 a^2} \paren {\int \frac {2 x \rd x} {x^2 \sqrt {x^2 - a^2} } } + C       | c = substituting for $z$ }} {{eqn | r = \frac {\sqrt {x^2 - a^2} } {2 a^2 x^2} + \frac 1 {2 a^2} \paren {\int \frac {\d x} {x \sqrt {x^2 - a^2} } } + C       | c = simplification }} {{eqn | r = \frac {\sqrt {x^2 - a^2} } {2 a^2 x^2} + \frac 1 {2 a^3} \arcsec \size {\frac x a} + C       | c = Primitive of $\dfrac 1 {x \sqrt {x^2 - a^2} }$ }} {{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 x \sqrt {x^2 - a^2} \rd x       | r = \int \frac {\sqrt z \sqrt {z - a^2} \rd z} {2 \sqrt z}       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \sqrt {z - a^2} \rd z       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \frac {2 \paren {\sqrt {z - a^2} }^3} 3 + C       | c = Primitive of $\sqrt {a x + b}$ }} {{eqn | r = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 + C       | c = substituting for $z$ and simplifying }} {{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 x^3 \sqrt {x^2 - a^2} \rd x       | r = \int \frac {z^{3/2} \sqrt {z - a^2} \rd z} {2 \sqrt z}       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int z \sqrt {z - a^2} \rd z       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \paren {\frac {2 \paren {3 z + 2 a^2} } {15} \paren {\sqrt {z - a^2} }^3 } + C       | c = Primitive of $x \sqrt {\paren {a x + b} }$ }} {{eqn | r = \frac {3 z + 2 a^2} {15} \paren {\sqrt {z - a^2} }^3 + C       | c = simplification }} {{eqn | r = \frac {3 x^2 + 2 a^2} {15} \paren {\sqrt {x^2 - a^2} }^3 + C       | c = substituting for $z$ }} {{eqn | r = \frac {3 x^2 - 3 a^2 + 5 a^2} {15} \paren {\sqrt {x^2 - a^2} }^3 + C       | c =  }} {{eqn | r = \frac {3 \paren {x^2 - a^2} \paren {\sqrt {a^2 - x^2} }^3 + 5 a^2 \paren {\sqrt {a^2 - x^2} }^3} {15} + C       | c =  }} {{eqn | r = \frac {\paren {\sqrt {x^2 - a^2} }^5} 5 + \frac {a^2 \paren {\sqrt {x^2 - a^2} }^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 {x^2 - a^2} } x \rd x       | r = \int \frac {\sqrt {z - a^2} \rd z} {2 \sqrt z \sqrt z}       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \frac {\sqrt {z - a^2} \rd z} z       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \paren {2 \sqrt {z - a^2} - a^2 \int \frac {\d z} {z \sqrt {z - a^2} } } + C       | c = Primitive of $\dfrac {\sqrt {a x + b} } x$ }} {{eqn | r = \sqrt {x^2 - a^2} - \frac {a^2} 2 \int \frac {2 x \rd x} {x^2 \sqrt {x^2 - a^2} } + C       | c = substituting for $z$ }} {{eqn | r = \sqrt {x^2 - a^2} - a^2 \int \frac {\d x} {x \sqrt {x^2 - a^2} } + C       | c = simplifying }} {{eqn | r = \sqrt {x^2 - a^2} - a^2 \paren {\frac 1 a \arcsec \size {\frac x a} } + C       | c = Primitive of $\dfrac 1 {x \sqrt {x^2 - a^2} }$ }} {{eqn | r = \sqrt {x^2 - a^2} - \frac 1 {2 a} \arcsec \size {\frac x a} + 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 {x^2 - a^2} } {x^2} \rd x       | r = \int \frac {\sqrt {z - a^2} \rd z} {2 z \sqrt z}       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \frac {\sqrt {z - a^2} \rd z} {z^{3/2} }       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \paren {\frac {-\sqrt {z - a^2} } {\frac 1 2 \sqrt z} + \frac 1 2 \int \frac {\d z} {\sqrt z \sqrt {z - a^2} } } + C       | c = Primitive of $\dfrac {\sqrt {a x + b} } {x^m}$ }} {{eqn | r = \frac {-\sqrt {x^2 - a^2} } x + \frac 1 2 \int \frac {2 x \rd x} {x \sqrt {x^2 - a^2} } + C       | c = substituting for $z$ }} {{eqn | r = \frac {-\sqrt {x^2 - a^2} } x + \int \frac {\d x} {\sqrt {x^2 - a^2} } + C       | c = simplifying }} {{eqn | r = \frac {-\sqrt {x^2 - a^2} } x + \ln \size {x + \sqrt {x^2 - a^2} } + C       | c = Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$ }} {{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 {\sqrt {x^2 - a^2} } {x^3} \rd x       | r = \int \frac {\sqrt {z - a^2} \rd z} {2 z^{3/2} \sqrt z}       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \frac {\sqrt {z - a^2} \rd z} {z^2}       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \paren {\frac {-\sqrt {z - a^2} } z + \frac 1 2 \int \frac {\d z} {z \sqrt {z - a^2} } } + C       | c = Primitive of $\dfrac {\sqrt {a x + b} } {x^m}$ }} {{eqn | r = \frac {-\sqrt {x^2 - a^2} } {2 x^2} + \frac 1 4 \int \frac {2 x \rd x} {x^2 \sqrt {x^2 - a^2} } + C       | c = substituting for $z$ }} {{eqn | r = \frac {-\sqrt {x^2 - a^2} } {2 x^2} + \frac 1 2 \int \frac {\d x} {x \sqrt {x^2 - a^2} } + C       | c = simplifying }} {{eqn | r = \frac {-\sqrt {x^2 - a^2} } {2 x^2} + \frac 1 2 \paren {\frac 1 a \arcsec \size {\frac x a} } + C       | c = Primitive of $\dfrac 1 {x \sqrt {x^2 - a^2} }$ }} {{eqn | r = \frac {-\sqrt {x^2 - a^2} } {2 x^2} + \frac 1 {2 a} \arcsec \size {\frac x a} + C       | c = simplifying }} {{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} {\paren {\sqrt {x^2 - a^2} }^3}       | r = \int \frac {\d z} {2 \sqrt z \paren {\sqrt {z - a^2} }^3}       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \frac {\mathrm d z} {\paren {z - a^2} \sqrt z \sqrt {z - a^2} }       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \paren {\frac {2 \sqrt z} {\paren {-a^2} \sqrt {z - a^2} } } + C       | c = Primitive of $\dfrac 1 {\paren {p x + q} \sqrt {\paren {a x + b} \paren {p x + q} } }$ }} {{eqn | r = \frac 1 2 \paren {\frac {2 x} {\paren {-a^2} \sqrt {x^2 - a^2} } } + C       | c = substituting for $z$ }} {{eqn | r = \frac {-x} {a^2 \sqrt {x^2 - a^2} } + C       | c = simplifying }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = z       | r = x^2 - a^2 }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \leadsto       | l = \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3}       | r = \int \frac {x \rd z} {2 x z^{3/2} }       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int z^{-3/2} \rd z       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \paren {\frac {-z^{1/2} } {\frac 1 2} } + C       | c = Primitive of Power }} {{eqn | r = \frac {-1} {\sqrt z} + C       | c = simplifying }} {{eqn | r = \frac {-1} {\sqrt {x^2 - a^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 {x^2 - a^2} }^3} }} {{eqn | ll= \leadsto       | l = v       | r = \frac {-1} {\sqrt {x^2 - a^2} }       | c = Primitive of $\dfrac x {\paren {\sqrt {x^2 - a^2} }^3}$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \frac {x^2 \rd x} {\paren {\sqrt {x^2 - a^2} }^3}       | r = \int x \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3}       | c =  }} {{eqn | r = x \frac {-1} {\sqrt {x^2 - a^2} } - \int \frac {-1} {\sqrt {x^2 - a^2} } 1 + C       | c = Integration by Parts }} {{eqn | r = \frac {-x} {\sqrt {x^2 - a^2} } + \int \frac 1 {\sqrt {x^2 - a^2} } + C       | c = simplifying }} {{eqn | r = \frac {-x} {\sqrt {x^2 - a^2} } + \ln \size {x + \sqrt {x^2 - a^2} } + C       | c = Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 - a^2} }^3}       | r = \int \frac {x \paren {x^2} \rd x} {\paren {\sqrt {x^2 - a^2} }^3}       | c =  }} {{eqn | r = \int \frac {x \paren {x^2 - a^2 + a^2} \rd x} {\paren {\sqrt {x^2 - a^2} }^3}       | c =  }} {{eqn | r = \int \frac {x \paren {x^2 - a^2} \rd x} {\paren {\sqrt {x^2 - a^2} }^3} + a^2 \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3}       | c = Linear Combination of Integrals }} {{eqn | r = \int \frac {x \rd x} {\sqrt {x^2 - a^2} } + a^2 \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3}       | c = simplifying }} {{eqn | r = \sqrt {x^2 - a^2} + a^2 \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3} + C       | c = Primitive of $\dfrac x {\sqrt {x^2 - a^2} }$ }} {{eqn | r = \sqrt {x^2 - a^2} + a^2 \frac {-1} {\sqrt {x^2 - a^2} } + C       | c = Primitive of $\dfrac x {\paren {\sqrt {x^2 - a^2} }^3}$ }} {{eqn | r = \sqrt {x^2 - a^2} - \frac {a^2} {\sqrt {x^2 - a^2} } + C       | c = simplifying }} {{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 \paren {\sqrt {x^2 - a^2} }^3}       | r = \int \frac {\d z} {2 \sqrt z \sqrt z \paren {\sqrt {z - a^2} }^3}       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \int \frac {\d z} {z \paren {\sqrt {z - a^2} }^3}       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \paren {\frac 2 {\paren {-a^2} \sqrt {z - a^2} } + \frac 1 {-a^2} \int \frac {\d z} {z \sqrt {z - a^2} } } + C       | c = Primitive of $\dfrac 1 {x \paren {\sqrt{a x + b} }^m}$ }} {{eqn | r = \frac {-1} {a^2 \sqrt {z - a^2} } - \frac 1 {2 a^2} \int \frac {\d z} {z \sqrt {z - a^2} } + C       | c = simplifying }} {{eqn | r = \frac {-1} {a^2 \sqrt {x^2 - a^2} } - \frac 1 {2 a^2} \int \frac {2 x \rd x} {x^2 \sqrt {x^2 - a^2} } + C       | c = substituting for $z$ }} {{eqn | r = \frac {-1} {a^2 \sqrt {x^2 - a^2} } - \frac 1 {a^2} \int \frac {\d x} {x \sqrt {x^2 - a^2} } + C       | c = simplifying }} {{eqn | r = \frac {-1} {a^2 \sqrt {x^2 + a^2} } - \frac 1 {a^3} \arcsec \size {\frac x a} + C       | c = Primitive of $\dfrac 1 {x \sqrt {x^2 - a^2} }$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\d x} {x^2 \paren {\sqrt {x^2 - a^2} }^3}       | r = \int \frac {a^2 \rd x} {a^2 x^2 \paren {\sqrt {x^2 - a^2} }^3}       | c =  }} {{eqn | r = \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x^2 \paren {\sqrt {x^2 - a^2} }^3}       | c =  }} {{eqn | r = \frac 1 {a^2} \int \frac {-\paren {x^2 - a^2} \rd x} {a^2 x^2 \paren {\sqrt {x^2 - a^2} }^3} + \frac 1 {a^2} \int \frac {x^2 \rd x} {x^2 \paren {\sqrt {x^2 - a^2} }^3}       | c = Linear Combination of Integrals }} {{eqn | r = \frac {-1} {a^2} \int \frac {\d x} {x^2 \sqrt {x^2 - a^2} } + \frac 1 {a^2} \int \frac {\d x} {\paren {\sqrt {x^2 - a^2} }^3}       | c = simplifying }} {{eqn | r = \frac {-1} {a^2} \frac {\sqrt {x^2 - a^2} } {a^2 x} + \frac 1 {a^2} \int \frac {\d x} {\paren {\sqrt {x^2 - a^2} }^3} + C       | c = Primitive of $\dfrac 1 {x^2 \sqrt {x^2 - a^2} }$ }} {{eqn | r = \frac {-\sqrt {x^2 - a^2} } {a^4 x} + \frac 1 {a^2} \frac {-x} {a^2 \sqrt {x^2 - a^2} } + C       | c = Primitive of $\dfrac 1 {\paren {\sqrt {x^2 - a^2} }^3}$ }} {{eqn | r = \frac {-\sqrt {x^2 - a^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 {\d x} {x^3 \paren {\sqrt {x^2 - a^2} }^3}       | r = \int \frac {a^2 \rd x} {a^2 x^3 \paren {\sqrt {x^2 - a^2} }^3}       | c =  }} {{eqn | r = \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x^3 \paren {\sqrt {x^2 - a^2} }^3}       | c =  }} {{eqn | r = \frac 1 {a^2} \int \frac {-\paren {x^2 - a^2} \rd x} {x^3 \paren {\sqrt {x^2 - a^2} }^3} + \frac 1 {a^2} \int \frac {x^2 \rd x} {x^3 \paren {\sqrt {x^2 - a^2} }^3}       | c = Linear Combination of Integrals }} {{eqn | r = \frac {-1} {a^2} \int \frac {\d x} {x^3 \sqrt {x^2 - a^2} } + \frac 1 {a^2} \int \frac {\d x} {x \paren {\sqrt {x^2 - a^2} }^3}       | c = simplifying }} {{eqn | r = \frac {-1} {a^2} \paren {\frac {\sqrt {x^2 - a^2} } {2 a^2 x^2} + \frac 1 {2 a^3} \arcsec \size {\frac x a} } + \frac 1 {a^2} \int \frac {\d x} {x \paren {\sqrt {x^2 - a^2} }^3} + C       | c = Primitive of $\dfrac 1 {x^3 \sqrt {x^2 - a^2} }$ }} {{eqn | r = \frac {-1} {a^2} \paren {\frac {\sqrt {x^2 - a^2} } {2 a^2 x^2} + \frac 1 {2 a^3} \arcsec \size {\frac x a} }       | c =  }} {{eqn | o =        | ro= +       | r = \frac 1 {a^2} \paren {\frac {-1} {a^2 \sqrt {x^2 - a^2} } - \frac 1 {a^3} \arcsec \size {\frac x a} } + C       | c = Primitive of $\dfrac 1 {x \paren {\sqrt {x^2 - a^2} }^3}$ }} {{eqn | r = \frac 1 {2 a^2 x^2 \sqrt {x^2 - a^2} } - \frac 3 {2 a^4 \sqrt {x^2 - a^2} } + \frac 3 {2 a^5} \arcsec \size {\frac x a} + 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 {x^2 - a^2} }^3 \rd x       | r = \int \frac {\paren {\sqrt {z - a^2} }^3} {2 \sqrt z} \rd x       | c = Integration by Substitution }} {{eqn | r = \frac {\sqrt z \paren {\sqrt {z - a^2} }^3} 4 - \frac {3 a^2} 8 \int \frac {\sqrt {z - a^2} } {\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 {z - a^2} }^3} 4 - \frac {3 a^2} 8 \paren {\sqrt z \sqrt {z - a^2} - \frac {a^2} 2 \int \frac {\d x} {\sqrt z \sqrt {z - a^2} } } + C       | c = Primitive of $\dfrac {\sqrt {p x + q} } {\sqrt{a x + b} }$ }} {{eqn | r = \frac {x \paren {\sqrt {x^2 - a^2} }^3} 4 - \frac {3 a^2} 8 x \sqrt {x^2 + a^2} + \frac {3 a^4} 8 \int \frac {\d x} {\sqrt {x^2 - a^2} } + C       | c = substituting for $z$ and simplifying }} {{eqn | r = \frac {x \paren {\sqrt {x^2 - a^2} }^3} 4 - \frac {3 a^2 x \sqrt {x^2 - a^2} } 8 + \frac {3 a^4} 8 \ln \size {x + \sqrt {x^2 - a^2} } + C       | c = Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$ }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = z       | r = x^2 - a^2 }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \leadsto       | l = \int x \paren {\sqrt {x^2 - a^2} }^3 \rd x       | r = \int \frac {z^{3/2} } 2 \rd z       | c = Integration by Substitution }} {{eqn | r = \frac 1 2 \frac {z^{5/2} } {\frac 5 2} + C       | c = Primitive of Power }} {{eqn | r = \frac {z^{5/2} } 5 + C       | c = simplifying }} {{eqn | r = \frac {\paren {\sqrt {x^2 - a^2} }^5} 5 + C       | c = substituting 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 x^2 \paren {\sqrt {x^2 - a^2} }^3 \rd x       | r = \int \frac {\sqrt z \paren {z - a^2}^{3/2} } 2 \rd z       | c = Integration by Substitution }} {{eqn | r = \frac {\sqrt z \paren {z - a^2}^{5/2} } 6 + \frac {a^2} {12} \int \frac {\paren {z - a^2}^{3/2} } {\sqrt z} \rd z + C       | c = Primitive of $\paren {p x + q}^n \sqrt{a x + b}$ }} {{eqn | r = \frac {x \paren {x^2 - a^2}^{5/2} } 6 + \frac {a^2} 6 \int \paren {x^2 - a^2}^{3/2} \rd x + C       | c = substituting for $z$ and simplifying }} {{eqn | r = \frac {x \paren {x^2 - a^2}^{5/2} } 6 }} {{eqn | o =       | ro= +       | r = \frac {a^2} 6 \paren {\frac {x \paren {\sqrt {x^2 - a^2} }^3} 4 - \frac {3 a^2 x \sqrt {x^2 - a^2} } 8 + \frac {3 a^4} 8 \ln \size {x + \sqrt {x^2 - a^2} } } + C       | c = Primitive of $\paren {\sqrt {x^2 - a^2} }^3$ }} {{eqn | r = \frac {x \paren {x^2 - a^2}^{5/2} } 6 + \frac {a^2 x \paren {\sqrt {x^2 - a^2} }^3} {24} }} {{eqn | o =       | ro= -       | r = \frac {a^4 x \sqrt {x^2 - a^2} } {16} + \frac {a^6} {16} \ln \size {x + \sqrt {x^2 - a^2} } + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int x^3 \paren {\sqrt {x^2 - a^2} }^3 \rd x       | r = \int x \paren {x^2} \paren {\sqrt {x^2 - a^2} }^3 \rd x       | c =  }} {{eqn | r = \int x \paren {x^2 - a^2 + a^2} \paren {\sqrt {x^2 - a^2} }^3 \rd x       | c = Primitive of Power }} {{eqn | r = \int x \paren {x^2 - a^2} \paren {\sqrt {x^2 - a^2} }^3 \rd x + a^2 \int x \paren {\sqrt {x^2 - a^2} }^3 \rd x       | c = Linear Combination of Integrals }} {{eqn | r = \int x \paren {\sqrt {x^2 - a^2} }^5 \rd x + a^2 \int x \paren {\sqrt {x^2 - a^2} }^3 \rd x       | c = simplifying }} {{end-eqn}} Let: {{begin-eqn}} {{eqn | l = z       | r = x^2 - a^2 }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\d x}       | r = 2 x       | c = Power Rule for Derivatives }} {{eqn | ll= \leadsto       | o =        | r = \int x \paren {\sqrt {x^2 - a^2} }^5 \rd x + a^2 \int x \paren {\sqrt {x^2 - a^2} }^3 \rd x       | c =  }} {{eqn | r = \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       | c = Integration by Substitution }} {{eqn | r =  \frac 1 2 \int z^{5/2} \rd z + \frac {a^2} 2 \int z^{3/2} \rd z       | c = simplifying }} {{eqn | r = \frac 1 2 \frac {z^{7/2} } {7/2} + \frac {a^2} 2 \frac {z^{5/2} } {5/2} + C       | c = Primitive of Power }} {{eqn | r = \frac {z^{7/2} } 7 + a^2 \frac {z^{5/2} } 5 + C       | c = simplifying }} {{eqn | r = \frac {\paren {\sqrt {x^2 - a^2} }^7} 7 + \frac {a^2 \paren {\sqrt {x^2 - a^2} }^5} 5 + C       | c = substituting 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 {\paren {\sqrt {x^2 - a^2} }^3} x \rd x       | r = \int \frac {\paren {\sqrt {z - a^2} }^3} {2 z} \rd z       | c = Integration by Substitution }} {{eqn | r = \frac {\paren {\sqrt {z - a^2} }^3} 3 - \frac {a^2} 2 \int \frac {\sqrt {z - a^2} } z \rd z       | c = Primitive of $\dfrac {\paren {\sqrt {a x + b} }^m} x$ }} {{eqn | r = \frac {\paren {\sqrt {z - a^2} }^3} 3 - \frac {a^2} 2 \paren {2 \sqrt {z - a^2} - a^2 \int \frac {\d z} {z \sqrt {z - a^2} } }       | c = Primitive of $\dfrac {\sqrt {a x + b} } x$ }} {{eqn | r = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 - a^2 \sqrt {x^2 - a^2} + a^4 \int \frac {\d x} {x \sqrt {x^2 - a^2} }       | c = substituting for $z$ and simplifying }} {{eqn | r = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 - a^2 \sqrt {x^2 - a^2} + a^3 \arcsec \size {\frac x a} + C       | c = Primitive of $\dfrac 1 {x \sqrt {x^2 - a^2} }$ }} {{end-eqn}} {{qed}}