nyuuzyou/wikis huggingface dataset:
Primitive of x squared over a squared minus x squared/Logarithm Form

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {a^2 - x^2} = -x + \frac a 2 \map \ln {\frac {a + x} {a - x} } + C$
for $x^2 < a^2$.
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.165$
Category:Primitive of x squared over a squared minus x squared
Primitive of x squared over a squared minus x squared squared

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\paren {a^2 - x^2}^2} = \frac x {2 \paren {a^2 - x^2} } - \frac 1 {4 a} \map \ln {\frac {a + x} {a - x} } + C$
for $x^2 < a^2$.
== Proof ==
== Also see ==
* Primitive of $\dfrac {x^2} {\paren {x^2 - a^2}^2}$
== Sources ==
* : $\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.172$
Category:Primitives involving a squared minus x squared
Primitive of x squared over a x + b

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {a x + b} = \frac {\paren {a x + b}^2} {2 a^3} - \frac {2 b \paren {a x + b} } {a^3} + \frac {b^2} {a^3} \ln \size {a x + b} + C$
== Proof 1 ==
== Proof 2 ==
== Examples ==
== Sources ==
* : $\S 14$: Integrals involving $a x + b$: $14.61$
* : $\S 17$: Tables of Special Indefinite Integrals: $(1)$ Integrals Involving $a x + b$: $17.1.3.$
Category:Primitive of x squared over a x + b‎
Category:Primitives involving a x + b
Primitive of x squared over a x + b/Examples/x^2 over x - 1

== Example of Use of Primitive of $\dfrac {x^2} {a x + b}$ ==
:$\ds \int \dfrac {x^2 \rd x} {x - 1} = \frac {x^2} 2 + x + \ln \size {x - 1} + C$
== Proof ==
From Primitive of $\dfrac {x^2} {a x + b}$:
Hence:
== Sources ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $21$.
Category:Examples of Use of Primitive of x squared over a x + b
Primitive of x squared over a x + b/Proof 2

== Theorem ==
== Proof ==
From Primitive of $x^m \paren {a x + b}^n$: Decrement of Power of $x$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x + b}^n \rd x$
Let $m = 2$ and $n = -1$.
Then:
Category:Primitive of x squared over a x + b‎
Primitive of x squared over a x + b cubed

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\paren {a x + b}^3} = \frac {2 b} {a^3 \paren {a x + b} } - \frac {b^2} {2 a^3 \paren {a x + b}^2} + \frac 1 {a^3} \ln \size {a x + b} + C$
== Proof ==
Put $u = a x + b$.
Then:
Then:
== Sources ==
* : $\S 14$: Integrals involving $a x + b$: $14.75$
* : $\S 17$: Tables of Special Indefinite Integrals: $(1)$ Integrals Involving $a x + b$: $17.1.13.$
Category:Primitives involving a x + b‎
Primitive of x squared over a x + b squared

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\paren {a x + b}^2} = \frac {a x + b} {a^3} - \frac {b^2} {a^3 \paren {a x + b} } - \frac {2 b} {a^3} \ln \size {a x + b} + C$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 14$: Integrals involving $a x + b$: $14.68$
* : $\S 17$: Tables of Special Indefinite Integrals: $(1)$ Integrals Involving $a x + b$: $17.1.8.$
Category:Primitive of x squared over a x + b squared
Category:Primitives involving a x + b
Primitive of x squared over a x + b squared by p x + q

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac {b^2} {\paren {b p - a q} a^2 \paren {a x + b} } + \frac 1 {\paren {b p - a q}^2} \paren {\frac {q^2} p \ln \size {p x + q} + \frac {b \paren {b p - 2 a q} } {a^2} \ln \size {a x + b} } + C$
== Proof ==
We have:
Therefore:
== Sources ==
* : $\S 14$: Integrals involving $a x + b$ and $p x + q$: $14.109$
* : $\S 17$: Tables of Special Indefinite Integrals: $(3)$ Integrals Involving $a x + b$ and $p x + q$: $17.3.5.$
Category:Primitive of x squared over a x + b squared by p x + q
Category:Primitives involving a x + b and p x + q
Primitive of x squared over a x + b squared by p x + q/Partial Fraction Expansion

== Lemma for Primitive of $\frac {x^2} {\paren {a x + b}^2 \paren {p x + q} }$ ==
:$\dfrac {x^2} {\paren {a x + b}^2 \paren {p x + q} } \equiv \dfrac {b \paren {b p - 2 a q} } {a \paren {b p - a q}^2 \paren {a x + b} } + \dfrac {-b^2} {a \paren {b p - a q} \paren {a x + b}^2} + \dfrac {q^2} {\paren {b p - a q}^2 \paren {p x + q} }$
== Proof ==
Setting $a x + b = 0$ in $(1)$:
Setting $p x + q = 0$ in $(1)$:
Equating $2$nd powers of $x$ in $(1)$:
Summarising:
Hence the result.
Category:Primitive of x squared over a x + b squared by p x + q
Category:Primitives involving a x + b and p x + q/Lemmata
Primitive of x squared over a x squared plus b x plus c

== Theorem ==
Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {a x^2 + b x + c} = \frac x a - \frac b {2 a^2} \ln \size {a x^2 + b x + c} + \frac {b^2 - 2 a c} {2 a^2} \int \frac {\d x} {a x^2 + b x + c}$
== Proof ==
== Examples ==
== Also see ==
* Primitive of $\dfrac 1 {a x^2 + b x + c}$
== Sources ==
* : $\S 14$: Integrals involving $a x^2 + b x + c$: $14.267$
Category:Primitive of x squared over a x squared plus b x plus c
Category:Primitives involving a x squared plus b x plus c
Primitive of x squared over a x squared plus b x plus c/Examples

== Examples of Use of Primitive of $\dfrac {x^2} {a x^2 + b x + c}$ ==
=== Primitive of $\dfrac {6 x^2 + 10 x + 5} {3 x^2 + 4 x + 2}$ ===
Category:Primitive of x squared over a x squared plus b x plus c
Primitive of x squared over a x squared plus b x plus c/Examples/6 x^2 + 10 x + 5 over 3 x^2 + 4 x + 2

== Example of Use of Primitive of $\dfrac {x^2} {a x^2 + b x + c}$ ==
:$\ds \int \dfrac {6 x^2 + 10 x + 5} {3 x^2 + 4 x + 2} \rd x = 2 x + \dfrac 1 3 \ln \size {3 x^3 + 4 x + 2} - \dfrac 1 {3 \sqrt 2} \arctan \dfrac {3 x + 2} {\sqrt 2} + C$
== Proof 1 ==
== Proof 2 ==
Category:Primitive of x squared over a x squared plus b x plus c/Examples/6 x^2 + 10 x + 5 over 3 x^2 + 4 x + 2
Category:Primitive of x squared over a x squared plus b x plus c
Primitive of x squared over a x squared plus b x plus c/Examples/6 x^2 + 10 x + 5 over 3 x^2 + 4 x + 2/Proof 1

== Example of Use of Primitive of $\dfrac {x^2} {a x^2 + b x + c}$ ==
== Proof ==
== Sources ==
* : $\text {II}$. Calculus: Integration: Algebraic Integration: Type $\text C$.
Category:Primitive of x squared over a x squared plus b x plus c/Examples/6 x^2 + 10 x + 5 over 3 x^2 + 4 x + 2
Primitive of x squared over a x squared plus b x plus c/Examples/6 x^2 + 10 x + 5 over 3 x^2 + 4 x + 2/Proof 2

== Example of Use of Primitive of $\dfrac {x^2} {a x^2 + b x + c}$ ==
== Proof ==
Category:Primitive of x squared over a x squared plus b x plus c/Examples/6 x^2 + 10 x + 5 over 3 x^2 + 4 x + 2
Primitive of x squared over square of a x squared plus b x plus c

== Theorem ==
Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {\paren {a x^2 + b x + c}^2} = \frac {\paren {b^2 - 2 a c} x + b c} {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {2 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $a x^2 + b x + c$: $14.274$
Category:Primitives involving a x squared plus b x plus c
Primitive of x squared over x cubed plus a cubed

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {x^3 + a^3} = \frac 1 3 \ln \size {x^3 + a^3} + C$ 
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 14$: Integrals involving $x^3 + a^3$: $14.301$
* : $\S 17$: Tables of Special Indefinite Integrals: $(14)$ Integrals Involving $x^3 + a^3$: $17.14.3.$
Category:Primitive of x squared over x cubed plus a cubed
Category:Primitives involving x cubed plus a cubed
Primitive of x squared over x cubed plus a cubed/Proof 2

== Theorem ==
== Proof ==
From Primitive of Power of x less one over Power of x plus Power of a:
:$\ds \int \frac {x^{n - 1} \rd x} {x^n + a^n} = \frac 1 n \ln \size {x^n + a^n} + C$
So:
directly.
Category:Primitive of x squared over x cubed plus a cubed
Primitive of x squared over x cubed plus a cubed squared

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\paren {x^3 + a^3}^2} = \frac {-1} {3 \paren {x^3 + a^3} } + C$ 
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $x^3 + a^3$: $14.306$
Category:Primitives involving x cubed plus a cubed
Primitive of x squared over x fourth minus a fourth

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {x^4 - a^4} = \frac 1 {4 a} \ln \size {\frac {x - a} {x + a} } + \frac 1 {2 a} \arctan \frac x a + C$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $x^4 \pm a^4$: $14.320$
Category:Primitives involving x to the fourth plus or minus a to the fourth
Primitive of x squared over x fourth plus a fourth

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {x^4 + a^4} = \frac 1 {4 a \sqrt 2} \map \ln {\frac {x^2 - a x \sqrt 2 + a^2} {x^2 + a x \sqrt 2 + a^2} } - \frac 1 {2 a \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} }$ 
== Proof ==
Then:
Similarly:
Thus:
== Sources ==
* : $\S 14$: Integrals involving $x^4 \pm a^4$: $14.313$
Category:Primitives involving x to the fourth plus or minus a to the fourth
Primitive of x squared over x fourth plus a fourth/Partial Fraction Expansion

== Lemma for Primitive of $\dfrac {x^2} {x^4 + a^4}$ ==
:$\dfrac {x^2} {x^4 + a^4} = \dfrac x {2 a \sqrt 2 \paren {x^2 - a x \sqrt 2 + a^2} } - \dfrac x {2 a \sqrt 2 \paren {x^2 + a x \sqrt 2 + a^2} }$
== Proof ==
Equating coefficients of $x^3$ in $(1)$:
Equating coefficients of $x^2$ in $(1)$:
Equating coefficients of $x$ in $(1)$:
Setting $x = 0$ in $(1)$:
Summarising:
Thus:
Category:Primitives involving x to the fourth plus or minus a to the fourth/Lemmata
Primitive of x squared over x squared minus a squared/Inverse Hyperbolic Cotangent Form

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {x^2 - a^2} = x - a \coth^{-1} \frac x a + C$
for $x^2 > a^2$.
== Proof ==
Let:
Category:Primitive of x squared over x squared minus a squared
Primitive of x squared over x squared minus a squared/Logarithm Form

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {x^2 - a^2} = x + \frac a 2 \map \ln {\frac {x - a} {x + a} } + C$
for $x^2 > a^2$.
== Proof ==
Let:
== Sources ==
* : $\S 14$: Integrals involving $x^2 - a^2$, $x^2 > a^2$: $14.146$
* : $\S 17$: Tables of Special Indefinite Integrals: $(7)$ Integrals Involving $x^2 - a^2$, $x^2 > a^2$: $17.7.3.$
Category:Primitive of x squared over x squared minus a squared
Primitive of x squared over x squared minus a squared squared

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\paren {x^2 - a^2}^2} = \frac {-x} {2 \paren {x^2 - a^2} } + \frac 1 {4 a} \map \ln {\frac {x - a} {x + a} } + C$
for $x^2 > a^2$.
== Proof ==
== Also see ==
* Primitive of $\dfrac {x^2} {\paren {a^2 - x^2}^2}$
== Sources ==
* : $\S 14$: Integrals involving $x^2 - a^2$, $x^2 > a^2$: $14.153$
* : $\S 17$: Tables of Special Indefinite Integrals: $(7)$ Integrals Involving $x^2 - a^2$, $x^2 > a^2$: $17.7.10.$
Category:Primitives involving x squared minus a squared
Primitive of x squared over x squared plus a squared

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {x^2 + a^2} = x - a \arctan {\frac x a} + C$
where $a$ is a non-zero constant.
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $x^2 + a^2$: $14.127$
* : $\S 17$: Tables of Special Indefinite Integrals: $(6)$ Integrals Involving $x^2 + a^2$: $17.6.3.$
Category:Primitives involving x squared plus a squared
Primitive of x squared over x squared plus a squared squared

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\paren {x^2 + a^2}^2} = \frac {-x} {2 \paren {x^2 + a^2} } + \frac 1 {2 a} \arctan \frac x a + C$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 14$: Integrals involving $x^2 + a^2$: $14.134$
* : $\S 17$: Tables of Special Indefinite Integrals: $(6)$ Integrals Involving $x^2 + a^2$: $17.6.10.$
Category:Primitive of x squared over x squared plus a squared squared
Category:Primitives involving x squared plus a squared
Primitives/Bernoulli Numbers/Mistake

== Source Work ==
:Chapter $17$: Tables of Special Indefinite Integrals
== Mistake ==
:Some integrals contain the Bernouilli numbers $B_n$ and the Euler numbers $E_n$ defined in Chapter $23$.
== Correction ==
The name is spelt .
== Sources ==
* : $\S 17$: Tables of Special Indefinite Integrals
Category:Mistakes/Mathematical Handbook of Formulas and Tables
Primitives involving Power of a squared minus x squared

== Theorem ==
This page gathers together the primitives of some expressions involving $\paren {x^2 - a^2}^n$.
=== Primitive of Reciprocal of $\paren {x^2 - a^2}^n$ ===
=== Primitive $x$ over $\paren {x^2 - a^2}^n$ ===
=== Primitive of Reciprocal of $x \paren {x^2 - a^2}^n$ ===
=== Primitive of $x^m$ over $\paren {x^2 - a^2}^2$ ===
=== Primitive of Reciprocal of $x^m \paren {x^2 - a^2}^n$ ===
== Also see ==
* Primitives involving $a^2 - x^2$ 
* Primitives involving $\paren {x^2 - a^2}^2$ 
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving a squared minus x squared
Primitives involving Power of x squared minus a squared

== Theorem ==
This page gathers together the primitives of some expressions involving $\left({x^2 - a^2}\right)^n$.
=== Primitive of Reciprocal of $\left({x^2 - a^2}\right)^n$ ===
=== Primitive $x$ over $\left({x^2 - a^2}\right)^n$ ===
=== Primitive of Reciprocal of $x \left({x^2 - a^2}\right)^n$ ===
=== Primitive of $x^m$ over $\left({x^2 - a^2}\right)^2$ ===
=== Primitive of Reciprocal of $x^m \left({x^2 - a^2}\right)^n$ ===
== Also see ==
* Primitives involving $x^2 - a^2$ 
* Primitives involving $\left({x^2 - a^2}\right)^2$ 
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving x squared minus a squared‎
Primitives involving Power of x squared plus a squared

== Theorem ==
This page gathers together the primitives of some expressions involving $\left({x^2 + a^2}\right)^n$.
=== Primitive of Reciprocal of $\left({x^2 + a^2}\right)^n$ ===
=== Primitive $x$ over $\left({x^2 + a^2}\right)^n$ ===
=== Primitive of Reciprocal of $x \left({x^2 + a^2}\right)^n$ ===
=== Primitive of $x^m$ over $\left({x^2 + a^2}\right)^2$ ===
=== Primitive of Reciprocal of $x^m \left({x^2 + a^2}\right)^n$ ===
== Also see ==
* Primitives involving $x^2 + a^2$ 
* Primitives involving $\left({x^2 + a^2}\right)^2$ 
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving x squared plus a squared‎
Primitives involving Root of a squared minus x squared

== Theorem ==
This page gathers together the primitives of some expressions involving $\sqrt {a^2 - x^2}$.
=== Primitive of Reciprocal of $\sqrt {a^2 - x^2}$ ===
=== Primitive of $x$ over $\sqrt {a^2 - x^2}$ ===
=== Primitive of $x^2$ over $\sqrt {a^2 - x^2}$ ===
=== Primitive of $x^3$ over $\sqrt {a^2 - x^2}$ ===
=== Primitive of Reciprocal of $x \sqrt {a^2 - x^2}$: $\sech^{-1}$ form ===
=== Primitive of Reciprocal of $x \sqrt {a^2 - x^2}$: Logarithm Form ===
=== Primitive of Reciprocal of $x^2 \sqrt {a^2 - x^2}$ ===
=== Primitive of Reciprocal of $x^3 \sqrt {a^2 - x^2}$ ===
=== Primitive of $\sqrt {a^2 - x^2}$ ===
=== Primitive of $x \sqrt {a^2 - x^2}$ ===
=== Primitive of $x^2 \sqrt {a^2 - x^2}$ ===
=== Primitive of $x^3 \sqrt {a^2 - x^2}$ ===
=== Primitive of $\sqrt {a^2 - x^2}$ over $x$ ===
=== Primitive of $\sqrt {a^2 - x^2}$ over $x^2$ ===
=== Primitive of $\sqrt {a^2 - x^2}$ over $x^3$ ===
== Also see ==
* Primitives involving $\paren {\sqrt{a^2 - x^2} }^3$ 
* Primitives involving $\sqrt {x^2 + a^2}$ 
* Primitives involving $\sqrt {x^2 - a^2}$ 
Category:Primitives
Category:Primitives involving Root of a squared minus x squared‎
Primitives involving Root of a squared minus x squared cubed

== Theorem ==
This page gathers together the primitives of some expressions involving $\left({\sqrt{a^2 - x^2} }\right)^3$.
=== Primitive of Reciprocal of $\left({\sqrt{a^2 - x^2} }\right)^3$ ===
=== Primitive of $x$ over $\left({\sqrt{a^2 - x^2} }\right)^3$ ===
=== Primitive of $x^2$ over $\left({\sqrt{a^2 - x^2} }\right)^3$ ===
=== Primitive of $x^3$ over $\left({\sqrt{a^2 - x^2} }\right)^3$ ===
=== Primitive of Reciprocal of $x \left({\sqrt{a^2 - x^2} }\right)^3$ ===
=== Primitive of Reciprocal of $x^2 \left({\sqrt{a^2 - x^2} }\right)^3$ ===
=== Primitive of Reciprocal of $x^3 \left({\sqrt{a^2 - x^2} }\right)^3$ ===
=== Primitive of $\left({\sqrt{a^2 - x^2} }\right)^3$ ===
=== Primitive of $x \left({\sqrt{a^2 - x^2} }\right)^3$ ===
=== Primitive of $x^2 \left({\sqrt{a^2 - x^2} }\right)^3$ ===
=== Primitive of $x^3 \left({\sqrt{a^2 - x^2} }\right)^3$ ===
=== Primitive of $\left({\sqrt{a^2 - x^2} }\right)^3$ over $x$ ===
=== Primitive of $\left({\sqrt{a^2 - x^2} }\right)^3$ over $x^2$ ===
=== Primitive of $\left({\sqrt{a^2 - x^2} }\right)^3$ over $x^3$ ===
== Also see ==
* Primitives involving $\left({\sqrt{x^2 + a^2} }\right)^3$ 
* Primitives involving $\left({\sqrt{x^2 - a^2} }\right)^3$ 
* Primitives involving $\sqrt{a^2 - x^2}$ 
Category:Primitives
Category:Primitives involving Root of a squared minus x squared‎
Primitives involving Root of x squared minus a squared

== Theorem ==
This page gathers together the primitives of some expressions involving $\sqrt{x^2 - a^2}$.
=== Primitive of Reciprocal of $\sqrt{x^2 - a^2}$: $\cosh^{-1}$ form ===
=== Primitive of Reciprocal of $\sqrt{x^2 - a^2}$: Logarithm Form ===
=== Primitive of $x$ over $\sqrt{x^2 - a^2}$ ===
=== Primitive of $x^2$ over $\sqrt{x^2 - a^2}$ ===
=== Primitive of $x^3$ over $\sqrt{x^2 - a^2}$ ===
=== Primitive of Reciprocal of $x \sqrt{x^2 - a^2}$ ===
=== Primitive of Reciprocal of $x^2 \sqrt{x^2 - a^2}$ ===
=== Primitive of Reciprocal of $x^3 \sqrt{x^2 - a^2}$ ===
=== Primitive of $\sqrt{x^2 - a^2}$ ===
=== Primitive of $x \sqrt{x^2 - a^2}$ ===
=== Primitive of $x^2 \sqrt{x^2 - a^2}$ ===
=== Primitive of $x^3 \sqrt{x^2 - a^2}$ ===
=== Primitive of $\sqrt{x^2 - a^2}$ over $x$ ===
=== Primitive of $\sqrt{x^2 - a^2}$ over $x^2$ ===
=== Primitive of $\sqrt{x^2 - a^2}$ over $x^3$ ===
== Also see ==
* Primitives involving $\left({\sqrt{x^2 - a^2} }\right)^3$ 
* Primitives involving $\sqrt{x^2 + a^2}$ 
* Primitives involving $\sqrt{a^2 - x^2}$ 
Category:Primitives
Category:Primitives involving Root of x squared minus a squared‎
Primitives involving Root of x squared minus a squared cubed

== Theorem ==
This page gathers together the primitives of some expressions involving $\left({\sqrt{x^2 - a^2} }\right)^3$.
=== Primitive of Reciprocal of $\left({\sqrt{x^2 - a^2} }\right)^3$ ===
=== Primitive of $x$ over $\left({\sqrt{x^2 - a^2} }\right)^3$ ===
=== Primitive of $x^2$ over $\left({\sqrt{x^2 - a^2} }\right)^3$ ===
=== Primitive of $x^3$ over $\left({\sqrt{x^2 - a^2} }\right)^3$ ===
=== Primitive of Reciprocal of $x \left({\sqrt{x^2 - a^2} }\right)^3$ ===
=== Primitive of Reciprocal of $x^2 \left({\sqrt{x^2 - a^2} }\right)^3$ ===
=== Primitive of Reciprocal of $x^3 \left({\sqrt{x^2 - a^2} }\right)^3$ ===
=== Primitive of $\left({\sqrt{x^2 - a^2} }\right)^3$ ===
=== Primitive of $x \left({\sqrt{x^2 - a^2} }\right)^3$ ===
=== Primitive of $x^2 \left({\sqrt{x^2 - a^2} }\right)^3$ ===
=== Primitive of $x^3 \left({\sqrt{x^2 - a^2} }\right)^3$ ===
=== Primitive of $\left({\sqrt{x^2 - a^2} }\right)^3$ over $x$ ===
=== Primitive of $\left({\sqrt{x^2 - a^2} }\right)^3$ over $x^2$ ===
=== Primitive of $\left({\sqrt{x^2 - a^2} }\right)^3$ over $x^3$ ===
== Also see ==
* Primitives involving $\left({\sqrt{x^2 + a^2} }\right)^3$ 
* Primitives involving $\left({\sqrt{a^2 - x^2} }\right)^3$ 
* Primitives involving $\sqrt{x^2 - a^2}$ 
Category:Primitives
Category:Primitives involving Root of x squared minus a squared‎
Primitives involving Root of x squared plus a squared

== Theorem ==
This page gathers together the primitives of some expressions involving $\sqrt {x^2 + a^2}$.
=== Primitive of Reciprocal of $\sqrt {x^2 + a^2}$: $\sinh^{-1}$ form ===
=== Primitive of Reciprocal of $\sqrt {x^2 + a^2}$: Logarithm Form ===
=== Primitive of $x$ over $\sqrt {x^2 + a^2}$ ===
=== Primitive of $x^2$ over $\sqrt {x^2 + a^2}$ ===
=== Primitive of $x^3$ over $\sqrt {x^2 + a^2}$ ===
=== Primitive of Reciprocal of $x \sqrt {x^2 + a^2}$: $\csch^{-1}$ form ===
=== Primitive of Reciprocal of $x \sqrt {x^2 + a^2}$: Logarithm Form ===
=== Primitive of Reciprocal of $x^2 \sqrt {x^2 + a^2}$ ===
=== Primitive of Reciprocal of $x^3 \sqrt {x^2 + a^2}$ ===
=== Primitive of $\sqrt {x^2 + a^2}$ ===
=== Primitive of $x \sqrt {x^2 + a^2}$ ===
=== Primitive of $x^2 \sqrt {x^2 + a^2}$ ===
=== Primitive of $x^3 \sqrt {x^2 + a^2}$ ===
=== Primitive of $\sqrt {x^2 + a^2}$ over $x$ ===
=== Primitive of $\sqrt {x^2 + a^2}$ over $x^2$ ===
=== Primitive of $\sqrt {x^2 + a^2}$ over $x^3$ ===
== Also see ==
* Primitives involving $\paren {\sqrt {x^2 + a^2} }^3$ 
* Primitives involving $\sqrt {x^2 - a^2}$ 
* Primitives involving $\sqrt {a^2 - x^2}$ 
Category:Primitives
Category:Primitives involving Root of x squared plus a squared‎
Primitives involving Root of x squared plus a squared cubed

== Theorem ==
This page gathers together the primitives of some expressions involving $\paren {\sqrt {x^2 + a^2} }^3$.
=== Primitive of Reciprocal of $\paren {\sqrt {x^2 + a^2} }^3$ ===
=== Primitive of $x$ over $\paren {\sqrt {x^2 + a^2} }^3$ ===
=== Primitive of $x^2$ over $\paren {\sqrt {x^2 + a^2} }^3$ ===
=== Primitive of $x^3$ over $\paren {\sqrt {x^2 + a^2} }^3$ ===
=== Primitive of Reciprocal of $x \paren {\sqrt {x^2 + a^2} }^3$ ===
=== Primitive of Reciprocal of $x^2 \paren {\sqrt {x^2 + a^2} }^3$ ===
=== Primitive of Reciprocal of $x^3 \paren {\sqrt {x^2 + a^2} }^3$ ===
=== Primitive of $\paren {\sqrt {x^2 + a^2} }^3$ ===
=== Primitive of $x \paren {\sqrt {x^2 + a^2} }^3$ ===
=== Primitive of $x^2 \paren {\sqrt {x^2 + a^2} }^3$ ===
=== Primitive of $x^3 \paren {\sqrt {x^2 + a^2} }^3$ ===
=== Primitive of $\paren {\sqrt {x^2 + a^2} }^3$ over $x$ ===
=== Primitive of $\paren {\sqrt {x^2 + a^2} }^3$ over $x^2$ ===
=== Primitive of $\paren {\sqrt {x^2 + a^2} }^3$ over $x^3$ ===
== Also see ==
* Primitives involving $\paren {\sqrt {x^2 - a^2} }^3$ 
* Primitives involving $\paren {\sqrt {a^2 - x^2} }^3$ 
* Primitives involving $\sqrt {x^2 + a^2}$ 
Category:Primitives
Category:Primitives involving Root of x squared plus a squared‎
Primitives involving a squared minus x squared

== Theorem ==
This page gathers together the primitives of some expressions involving $a^2 - x^2$.
=== Primitive of Reciprocal of $a^2 - x^2$: $\tanh^{-1}$ form ===
=== Primitive of Reciprocal of $a^2 - x^2$: Logarithm Form ===
=== Primitive $x$ over $a^2 - x^2$ ===
=== Primitive $x^2$ over $a^2 - x^2$ ===
=== Primitive $x^3$ over $a^2 - x^2$ ===
=== Primitive of Reciprocal of $x \left({a^2 - x^2}\right)$ ===
=== Primitive of Reciprocal of $x^2 \left({a^2 - x^2}\right)$ ===
=== Primitive of Reciprocal of $x^3 \left({a^2 - x^2}\right)$ ===
== Also see ==
* Primitives involving $\left({a^2 - x^2}\right)^2$ 
* Primitives involving $\left({a^2 - x^2}\right)^n$ 
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving a squared minus x squared‎
Primitives involving a squared minus x squared squared

== Theorem ==
This page gathers together the primitives of some expressions involving $\left({a^2 - x^2}\right)^2$.
=== Primitive of Reciprocal of $\left({a^2 - x^2}\right)^2$ ===
=== Primitive $x$ over $\left({a^2 - x^2}\right)^2$ ===
=== Primitive $x^2$ over $\left({a^2 - x^2}\right)^2$ ===
=== Primitive $x^3$ over $\left({a^2 - x^2}\right)^2$ ===
=== Primitive of Reciprocal of $x \left({a^2 - x^2}\right)^2$ ===
=== Primitive of Reciprocal of $x^2 \left({a^2 - x^2}\right)^2$ ===
=== Primitive of Reciprocal of $x^3 \left({a^2 - x^2}\right)^2$ ===
== Also see ==
* Primitives involving $a^2 - x^2$ 
* Primitives involving $\left({a^2 - x^2}\right)^n$ 
Category:Primitives
Category:Primitives of Rational Functions
Category:Primitives involving a squared minus x squared‎