hcju/proofwikiqa huggingface dataset:
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = \operatorname{csch}^{-1} \frac x a       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = \frac {-a} {\left\vert{x}\right\vert \sqrt{a^2 + x^2} }       | c = Derivative of $\operatorname{csch}^{-1} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm d x}       | r = 1       | c =  }} {{eqn | ll= \implies       | l = v       | r = x       | c = Primitive of Constant }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \operatorname{csch}^{-1} \frac x a \ \mathrm d x       | r = x \operatorname{csch}^{-1} \frac x a - \int x \left({\frac {-a} {\left\vert{x}\right\vert \sqrt{a^2 + x^2} } }\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = x \operatorname{csch}^{-1} \frac x a + a \int \frac {x \ \mathrm d x} {\left\vert{x}\right\vert \sqrt{a^2 + x^2} } + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = x \operatorname{csch}^{-1} \frac x a \begin{cases} \displaystyle \mathop + a \int \frac {\mathrm d x} {\sqrt{a^2 + x^2} } + C & : x > 0 \\ \displaystyle \mathop - a \int \frac {\mathrm d x} {\sqrt{a^2 + x^2} } + C & : x < 0\end{cases}       | c = Definition of Absolute Value }} {{eqn | r = \begin{cases} \displaystyle x \operatorname{csch}^{-1} \frac x a + a \sinh^{-1} \dfrac x a + C & : x > 0 \\ \displaystyle x \operatorname{csch}^{-1} \frac x a - a \sinh^{-1} \dfrac x a + C & : x < 0 \end{cases}       | c = Primitive of $\dfrac 1 {\sqrt{a^2 + x^2} }$ }} {{end-eqn}} {{qed}}
With a view to expressing the primitive 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 = \sinh^{-1} \frac x a       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \frac 1 {\sqrt {x^2 + a^2} }       | c = Derivative of $\sinh^{-1} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {x^2} 2       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x \sinh^{-1} \frac x a \rd x       | r = \frac {x^2} 2 \sinh^{-1} \frac x a - \int \frac {x^2} 2 \paren {\frac 1 {\sqrt {x^2 + a^2} } } \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^2} 2 \sinh^{-1} \frac x a - \frac 1 2 \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} } + C       | c = simplifying }} {{eqn | r = \frac {x^2} 2 \sinh^{-1} \frac x a - \frac 1 2 \paren {\frac {x \sqrt {x^2 + a^2} } 2 - \frac {a^2} 2 \sinh^{-1} \frac x a} + C       | c = Primitive of $\dfrac {x^2} {\sqrt {x^2 + a^2} }$ }} {{eqn | r = \paren {\frac {x^2} 2 + \frac {a^2} 4} \sinh^{-1} \frac x a - \frac {x \sqrt {x^2 + a^2} } 4 + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \sinh^{-1} \frac x a       | r = \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 + 1} }       | c = {{Defof|Inverse Hyperbolic Sine|subdef = Real|index = 2}} }} {{eqn | r = \map \ln {\frac x a + \sqrt {\frac {x^2 + a^2} {a^2} } }       | c =  }} {{eqn | r = \map \ln {\frac x a + \frac {\sqrt {x^2 + a^2} } a}       | c =  }} {{eqn | r = \map \ln {\frac {x + \sqrt {x^2 + a^2} } a}       | c =  }} {{eqn | r = \map \ln {x + \sqrt {x^2 + a^2} } - \ln a       | c = Difference of Logarithms }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cosh^{-1} \frac x a       | r = \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - 1} }       | c = {{Defof|Inverse Hyperbolic Cosine/Real|index = 2|Real Inverse Hyperbolic Cosine}} }} {{eqn | r = \map \ln {\frac x a + \sqrt {\frac {x^2 - a^2} {a^2} } }       | c =  }} {{eqn | r = \map \ln {\frac x a + \frac {\sqrt {x^2 - a^2} } a}       | c =  }} {{eqn | r = \map \ln {\frac {x + \sqrt {x^2 - a^2} } a}       | c =  }} {{eqn | r = \map \ln {x + \sqrt {x^2 - a^2} } - \ln a       | c = Difference of Logarithms }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tanh^{-1} \frac x a       | r = \frac 1 2 \map \ln {\frac {1 + \frac x a} {1 - \frac x a} }       | c = {{Defof|Inverse Hyperbolic Tangent|subdef = Real|index = 2}} }} {{eqn | r = \frac 1 2 \map \ln {\frac {a + x} {a - x} }       | c = multiplying top and bottom by $a$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \coth^{-1} \frac x a       | r = \frac 1 2 \map \ln {\frac {\frac x a + 1} {\frac x a - 1} }       | c = {{Defof|Inverse Hyperbolic Cotangent|subdef = Real|index = 2}} }} {{eqn | r = \frac 1 2 \map \ln {\frac {x + a} {x - a} }       | c = multiplying top and bottom by $a$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \operatorname{sech}^{-1} \frac x a       | r = \ln \left({\frac {1 + \sqrt{1 - \left({\frac x a}\right)^2} } {\frac x a} }\right)       | c = Definition of Inverse Hyperbolic secant }} {{eqn | r = \ln \left({\frac {a \left({1 + \sqrt{\dfrac {a^2 - x^2} {a^2} } }\right)} x}\right)       | c =  }} {{eqn | r = \ln \left({\frac {a \left({\dfrac {a + \sqrt{a^2 - x^2} } a}\right)} x}\right)       | c =  }} {{eqn | r = \ln \left({\frac {a + \sqrt{a^2 - x^2} } x}\right)       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sqrt {x^2 + a^2} \rd x       | r = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C       | c = Primitive of $\sqrt {x^2 + a^2}$ in $\sinh^{-1}$ form }} {{eqn | r = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \paren {\map \ln {x + \sqrt {x^2 + a^2} } - \ln a} + C       | c = $\sinh^{-1} \dfrac x a$ in Logarithm Form }} {{eqn | r = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } - \frac {a^2 \ln a} 2 + C       | c = simplifying }} {{eqn | r = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } + C       | c = subsuming $\dfrac {-a^2 \ln a} 2$ into the arbitrary constant }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = x       | r = a \sinh \theta }} {{eqn | n = 1       | ll= \leadsto       | l = \frac {\d x} {\d \theta}       | r = a \cosh \theta       | c = Derivative of Hyperbolic Sine }} {{end-eqn}} Also: {{begin-eqn}} {{eqn | l = x       | r = a \sinh \theta }} {{eqn | ll= \leadsto       | l = x^2 + a^2       | r = a^2 \sinh^2 \theta + a^2       | c =  }} {{eqn | r = a^2 \paren {\sinh^2 \theta + 1}       | c =  }} {{eqn | r = a^2 \cosh^2 \theta       | c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn | n = 2       | ll= \leadsto       | l = \sqrt {x^2 + a^2}       | r = a \cosh \theta       | c =  }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = x       | r = a \sinh \theta }} {{eqn | n = 3       | ll= \leadsto       | l = \theta       | r = \sinh^{-1} \frac x a       | c = {{Defof|Inverse Hyperbolic Sine/Real|index = 1|Real Inverse Hyperbolic Sine}} }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \int \sqrt {x^2 + a^2} \rd x       | r = \int \sqrt {x^2 + a^2} \, a \cosh \theta \rd \theta       | c = Integration by Substitution from $(1)$ }} {{eqn | r = \int a^2 \cosh^2 \theta \rd \theta       | c = substituting for $\sqrt {x^2 + a^2}$ from $(2)$ }} {{eqn | r = a^2 \int \cosh^2 \theta \rd \theta       | c = Primitive of Constant Multiple of Function }} {{eqn | r = a^2 \frac {\sinh \theta \cosh \theta + \theta} 2 + C       | c = Primitive of Square of Hyperbolic Cosine Function: Corollary }} {{eqn | r = \frac 1 2 a \sinh \theta a \cosh \theta + \frac {a^2 \theta} 2 + C       | c = rearranging }} {{eqn | r = \frac 1 2 x a \cosh \theta + \frac {a^2 \theta} 2 + C       | c = substituting $x = a \sinh \theta$ }} {{eqn | r = \frac 1 2 x \sqrt {x^2 + a^2} + \frac {a^2 \theta} 2 + C       | c = substituting $\sqrt {x^2 + a^2} = a \cosh \theta$ from $(2)$ }} {{eqn | r = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C       | c = substituting for $\theta$ from $(3)$ }} {{end-eqn}} {{qed}} {{proofread|I have never seen this offered as a possible solution, so is it possible it's not valid?}} Category:Primitive of Root of x squared plus a squared n0z8ljsog5exrhytu4dg6cplvl8togi
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       | c =  }} {{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 {\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^2 \rd x} {\sqrt {x^2 + a^2} }       | r = \int x \frac {x \rd x} {\sqrt {x^2 + a^2} }       | c =  }} {{eqn | r = x \sqrt {x^2 + a^2} - \int \sqrt {x^2 + a^2} \rd x       | c = Integration by Parts }} {{eqn | r = x \sqrt {x^2 + a^2} - \left({\frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a}\right) + C       | c = Primitive of $\sqrt {x^2 + a^2}$ }} {{eqn | r = \frac {x \sqrt {x^2 + a^2} } 2 - \frac {a^2} 2 \sinh^{-1} \frac x a + C       | c =  }} {{end-eqn}} {{qed}} Category:Primitive of x squared over Root of x squared plus a squared 8e58ojdzgfurhum4175e1f5pzzq6ch0
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       | c =  }} {{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 {\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^2 \rd x} {\sqrt {x^2 - a^2} }       | r = \int x \frac {x \rd x} {\sqrt {x^2 - a^2} }       | c =  }} {{eqn | r = x \sqrt {x^2 - a^2} - \int \sqrt {x^2 - a^2} \rd x       | c = Integration by Parts }} {{eqn | r = x \sqrt {x^2 - a^2} - \paren {\frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } } + C       | c = Primitive of $\sqrt {x^2 - a^2}$ }} {{eqn | r = \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C       | c =  }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = x       | r = a \cosh \theta }} {{eqn | n = 1       | ll= \leadsto       | l = \frac {\d x} {\d \theta}       | r = a \sinh \theta       | c = Derivative of Hyperbolic Cosine }} {{end-eqn}} Also: {{begin-eqn}} {{eqn | l = x       | r = a \cosh \theta }} {{eqn | ll= \leadsto       | l = x^2 - a^2       | r = a^2 \cosh^2 \theta - a^2       | c =  }} {{eqn | r = a^2 \paren {\cosh^2 \theta + 1}       | c =  }} {{eqn | r = a^2 \sinh^2 \theta       | c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn | n = 2       | ll= \leadsto       | l = \sqrt {x^2 - a^2}       | r = a \sinh \theta       | c =  }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = x       | r = a \cosh \theta }} {{eqn | n = 3       | ll= \leadsto       | l = \theta       | r = \cosh^{-1} \frac x a       | c = {{Defof|Inverse Hyperbolic Cosine/Real|index = 1|Real Inverse Hyperbolic Cosine}} }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | l = \int \sqrt {x^2 - a^2} \rd x       | r = \int \sqrt {x^2 - a^2} \, a \sinh \theta \rd \theta       | c = Integration by Substitution from $(1)$ }} {{eqn | r = \int a^2 \sinh^2 \theta \rd \theta       | c = substituting for $\sqrt {x^2 - a^2}$ from $(2)$ }} {{eqn | r = a^2 \int \sinh^2 \theta \rd \theta       | c = Primitive of Constant Multiple of Function }} {{eqn | r = a^2 \frac {\sinh \theta \cosh \theta - \theta} 2 + C       | c = Primitive of $\sinh^2 \theta$: Corollary }} {{eqn | r = \frac 1 2 a \sinh \theta a \cosh \theta - \frac {a^2 \theta} 2 + C       | c = rearranging }} {{eqn | r = \frac 1 2 x a \sinh \theta - \frac {a^2 \theta} 2 + C       | c = substituting $x = a \cosh \theta$ }} {{eqn | r = \frac 1 2 x \sqrt {x^2 - a^2} - \frac {a^2 \theta} 2 + C       | c = substituting $\sqrt {x^2 - a^2} = a \sinh \theta$ from $(2)$ }} {{eqn | r = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \cosh^{-1} \frac x a + C       | c = substituting for $\theta$ from $(3)$ }} {{end-eqn}} {{qed}} Category:Primitive of Root of x squared minus a squared djyd129bzuhwpu94dqdp4jxwkiutkh4
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       | c =  }} {{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 {\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^2 \rd x} {\sqrt {x^2 - a^2} }       | r = \int x \frac {x \rd x} {\sqrt {x^2 - a^2} }       | c =  }} {{eqn | r = x \sqrt {x^2 - a^2} - \int \sqrt {x^2 - a^2} \rd x       | c = Integration by Parts }} {{eqn | r = x \sqrt {x^2 - a^2} - \paren {\frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \cosh^{-1} \frac x a} + C       | c = Primitive of $\sqrt {x^2 - a^2}$ }} {{eqn | r = \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \cosh^{-1} \frac x a + C       | c = simplifying }} {{end-eqn}} Note that because: :$\cosh^{-1} \frac x a$ is defined for $x \ge a$ only and: :$\dfrac {x^2} {\sqrt {x^2 - a^2} } is not defined for $x = a$ $x$ is constrained as indicated. {{qed}}
With a view to expressing the primitive 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 = \cosh^{-1} \frac x a       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \frac 1 {\sqrt {x^2 - a^2} }       | c = Derivative of $\cosh^{-1} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {x^2} 2       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x \cosh^{-1} \frac x a \rd x       | r = \frac {x^2} 2 \cosh^{-1} \frac x a - \int \frac {x^2} 2 \paren {\frac 1 {\sqrt {x^2 - a^2} } } \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^2} 2 \cosh^{-1} \frac x a - \frac 1 2 \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} } + C       | c = simplifying }} {{eqn | r = \frac {x^2} 2 \cosh^{-1} \frac x a - \frac 1 2 \paren {\frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \cosh^{-1} \frac x a} + C       | c = Primitive of $\dfrac {x^2} {\sqrt {x^2 - a^2} }$ }} {{eqn | r = \paren {\frac {x^2} 2 - \frac {a^2} 4} \cosh^{-1} \frac x a - \frac {x \sqrt {x^2 - a^2} } 4 + C       | c = simplifying }} {{end-eqn}} {{qed}} {{finish|Analysis to be done for the negative case. The confusion here is that $\cosh^{-1} \frac x a$ is ''defined'' as being positive.}}
Let: {{begin-eqn}} {{eqn | l = \int \frac {x^2 \ \mathrm d x} {x^2 - a^2}       | r = \int \frac {x^2 - a^2 + a^2} {x^2 - a^2} \ \mathrm d x       | c =  }} {{eqn | r = \int \frac {x^2 - a^2} {x^2 - a^2} \ \mathrm d x + \int \frac {a^2} {x^2 - a^2} \ \mathrm d x       | c =  }} {{eqn | r = \int \mathrm d x + a^2 \int \frac {\mathrm d x} {x^2 - a^2}       | c = Primitive of Constant Multiple of Function }} {{eqn | r = x + a^2 \int \frac {\mathrm d x} {x^2 - a^2} + C       | c = Primitive of Constant }} {{eqn | r = x + a^2 \left({-\frac 1 a \coth^{-1} {\frac x a} }\right) + C       | c = Primitive of Reciprocal of $x^2 - a^2$: $\coth^{-1}$ form }} {{eqn | r = x - a \coth^{-1} {\frac x a} + C       | c = simplifying }} {{end-eqn}} {{qed}} Category:Primitive of x squared over x squared minus a squared l98lpnwxoktlylc6pngqudn6q75wtyc
{{begin-eqn}} {{eqn | l = \int \frac {x^2 \rd x} {a^2 - x^2}       | r = \int \frac {x^2 - a^2 + a^2} {a^2 - x^2} \rd x       | c =  }} {{eqn | r = \int \frac {- \left({a^2 - x^2}\right)} {a^2 - x^2} \rd x + \int \frac {a^2} {a^2 - x^2} \rd x       | c = Linear Combination of Integrals }} {{eqn | r = - \int \d x + a^2 \int \frac {\d x} {a^2 - x^2}       | c = Linear Combination of Integrals }} {{eqn | r = -x + a^2 \int \frac {\d x} {a^2 - x^2} + C       | c = Primitive of Constant }} {{eqn | r = -x + a^2 \paren {\frac 1 {2 a} \map \ln {\frac {a + x} {a - x} } } + C       | c = Primitive of $\dfrac 1 {a^2 - x^2}$: Logarithm Form }} {{eqn | r = -x + \frac a 2 \map \ln {\frac {a + x} {a - x} } + C       | c = simplifying }} {{end-eqn}} {{qed}}
Let: {{begin-eqn}} {{eqn | l = \int \frac {x^2 \rd x} {a^2 - x^2}       | r = \int \frac {x^2 - a^2 + a^2} {a^2 - x^2} \rd x       | c =  }} {{eqn | r = \int \frac {- \left({a^2 - x^2}\right)} {a^2 - x^2} \rd x + \int \frac {a^2} {a^2 - x^2} \rd x       | c = Linear Combination of Integrals }} {{eqn | r = - \int \rd x + a^2 \int \frac {\rd x} {a^2 - x^2}       | c = Linear Combination of Integrals }} {{eqn | r = -x + a^2 \int \frac {\rd x} {a^2 - x^2} + C       | c = Primitive of Constant }} {{eqn | r = -x + a^2 \left({\frac 1 a \tanh^{-1} \frac x a}\right) + C       | c = Primitive of Reciprocal of $a^2 - x^2$: $\tanh^{-1}$ form }} {{eqn | r = -x + a \tanh^{-1} \frac x a + C       | c = simplifying }} {{end-eqn}} {{qed}} Category:Primitive of x squared over a squared minus x squared pg0zh6tf9dnyu8rdcx9ic2og1wdag3n
With a view to expressing the primitive 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 = \tanh^{-1} \frac x a       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \frac a {a^2 - x^2}       | c = Derivative of $\tanh^{-1} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm d x}       | r = x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {x^2} 2       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x \tanh^{-1} \frac x a \rd x       | r = \frac {x^2} 2 \tanh^{-1} \frac x a - \int \frac {x^2} 2 \paren {\frac a {a^2 - x^2} } \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^2} 2 \tanh^{-1} \frac x a - \frac a 2 \int \frac {x^2 \rd x} {a^2 - x^2} + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac {x^2} 2 \tanh^{-1} \frac x a - \frac a 2 \paren {-x + a \tanh^{-1} \frac x a} + C       | c = Primitive of $\dfrac {x^2} {a^2 - x^2}$ }} {{eqn | r = \frac {a x} 2 + \frac {x^2 - a^2} 2 \tanh^{-1} \frac x a + C       | c = simplifying }} {{end-eqn}} {{qed}}
With a view to expressing the primitive 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 = \coth^{-1} \frac x a       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \frac a {a^2 - x^2}       | c = Derivative of $\coth^{-1} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {x^2} 2       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x \coth^{-1} \frac x a \rd x       | r = \frac {x^2} 2 \coth^{-1} \frac x a - \int \frac {x^2} 2 \paren {\frac a {a^2 - x^2} } \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^2} 2 \coth^{-1} \frac x a + \frac a 2 \int \frac {x^2 \rd x} {x^2 - a^2} + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac {x^2} 2 \coth^{-1} \frac x a + \frac a 2 \paren {x - a \coth^{-1} \frac x a} + C       | c = Primitive of $\dfrac {x^2} {x^2 - a^2}$ }} {{eqn | r = \frac {a x} 2 + \frac {x^2 - a^2} 2 \coth^{-1} \frac x a + C       | c = simplifying }} {{end-eqn}} {{qed}}
With a view to expressing the primitive 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 = \sinh^{-1} \frac x a       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \frac 1 {\sqrt {x^2 + a^2} }       | c = Derivative of $\sinh^{-1} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = x^2       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {x^3} 3       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x^2 \sinh^{-1} \frac x a \rd x       | r = \frac {x^3} 3 \sinh^{-1} \frac x a - \int \frac {x^3} 3 \paren {\frac 1 {\sqrt {x^2 + a^2} } } \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^3} 3 \sinh^{-1} \frac x a - \frac 1 3 \int \frac {x^3 \rd x} {\sqrt {x^2 + a^2} } + C       | c = simplifying }} {{eqn | r = \frac {x^3} 3 \sinh^{-1} \frac x a - \frac 1 3 \paren {\frac {\paren {\sqrt {x^2 + a^2} }^3} 3 - a^2 \sqrt {x^2 + a^2} } + C       | c = Primitive of $\dfrac {x^3} {\sqrt {x^2 + a^2} }$ }} {{eqn | r = \frac {x^3} 3 \sinh^{-1} \frac x a + \frac {\paren {2 a^2 - x^2} \sqrt {x^2 + a^2} } 9 + C       | c = simplifying }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = \cosh^{-1} \frac x a       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = \frac 1 {\sqrt {x^2 - a^2} }       | c = Derivative of $\cosh^{-1} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm d x}       | r = x^2       | c =  }} {{eqn | ll= \implies       | l = v       | r = \frac {x^3} 3       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x^2 \cosh^{-1} \frac x a \ \mathrm d x       | r = \frac {x^3} 3 \cosh^{-1} \frac x a - \int \frac {x^3} 3 \left({\frac 1 {\sqrt {x^2 - a^2} } }\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^3} 3 \cosh^{-1} \frac x a - \frac 1 3 \int \frac {x^3 \ \mathrm d x} {\sqrt {x^2 - a^2} } + C       | c = simplifying }} {{eqn | r = \frac {x^3} 3 \cosh^{-1} \frac x a - \frac 1 3 \left({\frac {\left({\sqrt {x^2 - a^2} }\right)^3} 3 + a^2 \sqrt {x^2 - a^2} }\right) + C       | c = Primitive of $\dfrac {x^3} {\sqrt {x^2 - a^2} }$ }} {{eqn | r = \frac {x^3} 3 \cosh^{-1} \frac x a - \frac {\left({x^2 + 2 a^2}\right) \sqrt {x^2 - a^2} } 9 + C       | c = simplifying }} {{end-eqn}} {{qed}} {{finish|Analysis to be done for the negative case. The confusion here is that $\cosh^{-1} \frac x a$ is ''defined'' as being positive.}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = \tanh^{-1} \frac x a       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = \frac a {a^2 - x^2}       | c = Derivative of $\tanh^{-1} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm d x}       | r = x^2       | c =  }} {{eqn | ll= \implies       | l = v       | r = \frac {x^3} 3       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x^2 \tanh^{-1} \frac x a \ \mathrm d x       | r = \frac {x^3} 3 \tanh^{-1} \frac x a - \int \frac {x^3} 3 \left({\frac a {a^2 - x^2} }\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^3} 3 \tanh^{-1} \frac x a - \frac a 3 \int \frac {x^3 \ \mathrm d x} {a^2 - x^2} + C       | c = simplifying }} {{eqn | r = \frac {x^3} 3 \tanh^{-1} \frac x a - \frac a 3 \left({-\frac {x^2} 2 - \frac {a^2} 2 \ln \left({a^2 - x^2}\right) }\right) + C       | c = Primitive of $\dfrac {x^3} {a^2 - x^2}$ }} {{eqn | r = \frac {a x^2} 6 + \frac {x^3} 3 \tanh^{-1} \frac x a + \frac {a^3} 6 \ln \left({a^2 - x^2}\right) + C       | c = simplifying }} {{end-eqn}} {{qed}}