nyuuzyou/wikis huggingface dataset:
Primitive of x squared by Arcsine of x over a

== Theorem ==
:$\ds \int x^2 \arcsin \frac x a \rd x = \frac {x^3} 3 \arcsin \frac x a + \frac {\paren {x^2 + 2 a^2} \sqrt {a^2 - x^2} } 9 + C$
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $x^2 \arccos \dfrac x a$
* Primitive of $x^2 \arctan \dfrac x a$
* Primitive of $x^2 \arccot \dfrac x a$
* Primitive of $x^2 \arcsec \dfrac x a$
* Primitive of $x^2 \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.473$
Category:Primitives involving Inverse Sine Function
Primitive of x squared by Arctangent of x over a

== Theorem ==
:$\ds \int x^2 \arctan \frac x a \rd x = \frac {x^3} 3 \arctan \frac x a - \frac {a x^2} 6 + \frac {a^3} 6 \map \ln {x^2 + a^2} + C$
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $x^2 \arcsin \dfrac x a$
* Primitive of $x^2 \arccos \dfrac x a$
* Primitive of $x^2 \arccot \dfrac x a$
* Primitive of $x^2 \arcsec \dfrac x a$
* Primitive of $x^2 \arccsc \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.485$
Category:Primitives involving Inverse Tangent Function
Primitive of x squared by Cosine of a x

== Theorem ==
:$\ds \int x^2 \cos a x \rd x = \frac {2 x \cos a x} {a^2} + \paren {\frac {x^2} a - \frac 2 {a^3} } \sin a x + C$
where $C$ is an arbitrary constant.
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $x^2 \sin a x$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.371$
Category:Primitives involving Cosine Function
Primitive of x squared by Cosine of x

== Theorem ==
:$\ds \int x^2 \cos x \rd x = x^2 \sin x + 2 x \cos x + 2 \sin x + C$
== Proof ==
From Primitive of $x^2 \cos a x$:
The result follows on setting $a = 1$.
== Sources ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $23$.
Category:Primitives involving Cosine Function
Primitive of x squared by Exponential of a x

== Theorem ==
:$\ds \int x^2 e^{a x} \rd x = \frac {e^{a x} } a \paren {x^2 - \frac {2 x} a + \frac 2 {a^2} } + C$
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Examples ==
== Also see ==
* Primitive of $x^n e^{a x}$
* Primitive of $x e^{a x} $
== Sources ==
* : $\S 14$: Integrals involving $e^{a x}$: $14.511$
Category:Primitive of x squared by Exponential of a x
Category:Primitives involving Exponential Function
Primitive of x squared by Exponential of a x/Examples/x squared by e^-x

== Theorem ==
:$\ds \int x^2 e^{-x} \rd x = -e^{-x} \paren {x^2 + 2 x + 2} + C$
== Proof ==
From Primitive of $x^2 e^{a x}$:
So:
Category:Primitives involving Cosine Function
Primitive of x squared by Exponential of x/Proof 2

== Theorem ==
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Sources ==
* : $\text {II}$. Calculus: Integration: Integration by Parts: Example
Category:Primitive of x squared by Exponential of x
Primitive of x squared by Hyperbolic Cosine of a x

== Theorem ==
:$\ds \int x^2 \cosh a x \rd x = \paren {\frac {x^2} a + \frac 2 {a^3} } \sinh a x - \frac {2 x \cosh a x} {a^2} + C$
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $x^2 \sinh a x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.564$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of x squared by Hyperbolic Sine of a x

== Theorem ==
:$\ds \int x^2 \sinh a x \rd x = \paren {\frac {x^2} a + \frac 2 {a^3} } \cosh a x - \frac {2 x \sinh a x} {a^2} + C$
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $x^2 \cosh a x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.542$
Category:Primitives involving Hyperbolic Sine Function
Primitive of x squared by Inverse Hyperbolic Cosine of x over a

== Theorem ==
:$\ds \int x^2 \arcosh \frac x a \rd x = \dfrac {x^3} 3 \arcosh \dfrac x a - \dfrac {\paren {x^2 + 2 a^2} \sqrt {x^2 - a^2} } 9 + C$
where $\arcosh$ denotes the real area hyperbolic cosine.
=== Corollary ===
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $x^2 \arsinh \dfrac x a$
* Primitive of $x^2 \artanh \dfrac x a$
* Primitive of $x^2 \arcoth \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.653$
Category:Primitives involving Inverse Hyperbolic Cosine Function
Category:Primitive of x squared by Inverse Hyperbolic Cosine of x over a
Primitive of x squared by Inverse Hyperbolic Cosine of x over a/Corollary

== Theorem ==
:$\ds \int x^2 \paren {-\cosh^{-1} \dfrac x a} \rd x = \dfrac {x^3} 3 \paren {-\cosh^{-1} \dfrac x a} - \dfrac {\paren {x^2 + 2 a^2} \sqrt {x^2 - a^2} } 9 + C$
where $-\cosh^{-1}$ denotes the negative branch of the real inverse hyperbolic cosine multifunction.
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.653$
Category:Primitive of x squared by Inverse Hyperbolic Cosine of x over a
Primitive of x squared by Inverse Hyperbolic Cotangent of x over a

== Theorem ==
:$\ds \int x^2 \arcoth \frac x a \rd x = \frac {a x^2} 6 + \frac {x^3} 3 \arcoth \frac x a + \frac {a^3} 6 \map \ln {x^2 - a^2} + C$
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $x^2 \arsinh \dfrac x a$
* Primitive of $x^2 \arcosh \dfrac x a$
* Primitive of $x^2 \artanh \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.663$
Category:Primitives involving Inverse Hyperbolic Cotangent Function
Primitive of x squared by Inverse Hyperbolic Sine of x over a

== Theorem ==
:$\ds \int x^2 \arsinh \frac x a \rd x = \frac {x^3} 3 \arsinh \frac x a + \frac {\paren {2 a^2 - x^2} \sqrt {x^2 + a^2} } 9 + C$
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $x^2 \arcosh \dfrac x a$
* Primitive of $x^2 \artanh \dfrac x a$
* Primitive of $x^2 \arcoth \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.648$
Category:Primitives involving Inverse Hyperbolic Sine Function
Primitive of x squared by Inverse Hyperbolic Tangent of x over a

== Theorem ==
:$\ds \int x^2 \artanh \frac x a \rd x = \frac {a x^2} 6 + \frac {x^3} 3 \artanh \frac x a + \frac {a^3} 6 \map \ln {a^2 - x^2} + C$
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $x^2 \arsinh \dfrac x a$
* Primitive of $x^2 \arcosh \dfrac x a$
* Primitive of $x^2 \arcoth \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.658$
Category:Primitives involving Inverse Hyperbolic Tangent Function
Primitive of x squared by Power of Root of a x + b

== Theorem ==
:$\ds \int x^2 \paren {\sqrt {a x + b} }^m \rd x = \frac {2 \paren {\sqrt {a x + b} }^{m + 6} } {a^3 \paren {m + 6} } - \frac {4 b \paren {\sqrt {a x + b} }^{m + 4} } {a^3 \paren {m + 4} } + \frac {2 b^2 \paren {\sqrt {a x + b} }^{m + 2} } {a^3 \paren {m + 2} } + C$
== Proof ==
Let:
Then:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x + b}$: $14.101$
* : $\S 17$: Tables of Special Indefinite Integrals: $(2)$ Integrals Involving $\sqrt {a x + b}$: $17.2.18.$
Category:Primitives involving Root of a x + b
Primitive of x squared by Power of a x + b

== Theorem ==
:$\ds \int x^2 \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 3} } {\paren {n + 3} a^3} - \frac {2 b \paren {a x + b}^{n + 2} } {\paren {n + 2} a^3} + \frac {b^2 \paren {a x + b}^{n + 1} } {\paren {n + 1} a^3} + C$
where $n \notin \set {-1, -2, -3}$.
== Proof ==
Let $u = a x + b$.
Then:
Then:
== Also see ==
* Primitive of $x^2$ over $a x + b$ for the case when $n = -1$
* Primitive of $x^2$ over $\paren {a x + b}^2$ for the case when $n = -2$
* Primitive of $x^2$ over $\paren {a x + b}^3$ for the case when $n = -3$
== Sources ==
* : $\S 14$: Integrals involving $a x + b$: $14.82$
* : $\S 17$: Tables of Special Indefinite Integrals: $(1)$ Integrals Involving $a x + b$: $17.1.16.$
Category:Primitives involving a x + b‎
Primitive of x squared by Root of a squared minus x squared

== Theorem ==
:$\ds \int x^2 \sqrt {a^2 - x^2} \rd x = -\frac {x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {a^2 x \sqrt {a^2 - x^2} } 8 + \frac {a^4} 8 \arcsin \frac x a + C$
== Proof 1 ==
== Proof 2 ==
== Also presented as ==
== Also see ==
* Primitive of $x^2 \sqrt {x^2 + a^2}$
* Primitive of $x^2 \sqrt {x^2 - a^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a^2 - x^2}$: $14.246$
Category:Primitive of x squared by Root of a squared minus x squared
Category:Primitives involving Root of a squared minus x squared
Primitive of x squared by Root of a squared minus x squared/Also presented as

== Primitive of $x^2 \sqrt {a^2 - x^2}$: Also presented as ==
This result is also seen presented in the form:
:$\ds \int x^2 \sqrt {a^2 - x^2} \rd x = \frac {a^4} 8 \arcsin \frac x a - \frac {x \sqrt {a^2 - x^2} \paren {a^2 - 2 x^2} } 8 + C$
== Proof ==
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $30$.
Category:Primitive of x squared by Root of a squared minus x squared
Primitive of x squared by Root of a squared minus x squared/Proof 1

== Theorem ==
== Proof ==
Let us assume that $a > 0$.
Let:
Hence:
Recall:
Hence:
== Also see ==
* Primitive of $x^2 \sqrt {x^2 + a^2}$
* Primitive of $x^2 \sqrt {x^2 - a^2}$
== Sources ==
* 
Category:Primitive of x squared by Root of a squared minus x squared
Primitive of x squared by Root of a squared minus x squared/Proof 2

== Theorem ==
== Proof ==
Recall:
Then:
Then:
Recall:
Then:
== Also see ==
* Primitive of $x^2 \sqrt {x^2 + a^2}$
* Primitive of $x^2 \sqrt {x^2 - a^2}$
Category:Primitive of x squared by Root of a squared minus x squared
Primitive of x squared by Root of a squared minus x squared cubed

== Theorem ==
:$\ds \int x^2 \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {-x \paren {\sqrt {a^2 - x^2} }^5} 6 + \frac {a^2 x \paren {\sqrt {a^2 - x^2} }^3} {24} + \frac {a^4 x \sqrt {a^2 - x^2} } {16} + \frac {a^6} {16} \arcsin \frac x a + C$
== Proof ==
Let:
== Also see ==
* Primitive of $x^2 \paren {\sqrt {x^2 + a^2} }^3$
* Primitive of $x^2 \paren {\sqrt {x^2 - a^2} }^3$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a^2 - x^2}$: $14.260$
Category:Primitives involving Root of a squared minus x squared
Primitive of x squared by Root of a x + b

== Theorem ==
:$\ds \int x^2 \sqrt {a x + b} \rd x = \frac {2 \paren {15 a^2 x^2 - 12 a b x + 8 b^2} } {105 a^3} \sqrt {\paren {a x + b}^3} + C$
== Proof ==
Let:
Then:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x + b}$: $14.91$
* : $\S 17$: Tables of Special Indefinite Integrals: $(2)$ Integrals Involving $\sqrt {a x + b}$: $17.2.8.$
Category:Primitives involving Root of a x + b
Primitive of x squared by Root of a x squared plus b x plus c

== Theorem ==
Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int x^2 \sqrt {a x^2 + b x + c} \rd x = \frac {6 a x - 5 b} {24 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {5 b^2 - 4 a c} {16 a^2} \int \sqrt {a x^2 + b x + c} \rd x$ 
== Proof ==
Let:
With a view to expressing the primitive $\ds \int x \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x$ in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
Now consider:
Thus:
Hence:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x^2 + bx + c}$: $14.287$
Category:Primitives involving Root of a x squared plus b x plus c
Primitive of x squared by Root of x squared minus a squared

== Theorem ==
:$\ds \int x^2 \sqrt {x^2 - a^2} \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^3} 4 + \frac {a^2 x \sqrt {x^2 - a^2} } 8 - \frac {a^4} 8 \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $\size x \ge a$.
== Proof ==
We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 \ge a^2$, that is, either:
:$x \ge a$
or:
:$x \le -a$
where it is assumed that $a > 0$.
First let $x \ge a$.
Let:
Now suppose $x \le -a$.
Let $z = -x$.
Then:
:$\d x = -\d z$
and we then have:
== Also presented as ==
== Also see ==
* Primitive of $x^2 \sqrt {x^2 + a^2}$
* Primitive of $x^2 \sqrt {a^2 - x^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.218$
Category:Primitive of x squared by Root of x squared minus a squared
Category:Primitives involving Root of x squared minus a squared
Primitive of x squared by Root of x squared minus a squared/Also presented as

== Primitive of $x^2 \sqrt {x^2 - a^2}$: Also presented as ==
This result is also seen presented in the form:
:$\ds \int x^2 \sqrt {x^2 - a^2} \rd x = \frac x 8 \paren {2 x^2 - a^2} \sqrt {x^2 - a^2} - \frac {a^4} 8 \arcosh \dfrac x a + C$
== Proof ==
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $41$.
Category:Primitive of x squared by Root of x squared minus a squared
Primitive of x squared by Root of x squared minus a squared cubed

== Theorem ==
:$\ds \int x^2 \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^5} 6 + \frac {a^2 x \paren {\sqrt {x^2 - a^2} }^3} {24} - \frac {a^4 x \sqrt {x^2 - a^2} } {16} + \frac {a^6} {16} \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $\size x \ge a$.
== Proof ==
Let:
== Also see ==
* Primitive of $x^2 \paren {\sqrt {x^2 + a^2} }^3$
* Primitive of $x^2 \paren {\sqrt {a^2 - z^2} }^3$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.232$
Category:Primitives involving Root of x squared minus a squared
Primitive of x squared by Root of x squared plus a squared

== Theorem ==
:$\ds \int x^2 \sqrt {x^2 + a^2} \rd x = \frac {x \paren {\sqrt {x^2 + a^2} }^3} 4 - \frac {a^2 x \sqrt {x^2 + a^2} } 8 - \frac {a^4} 8 \map \ln {x + \sqrt {x^2 + a^2} } + C$
== Proof ==
Let:
== Also presented as ==
== Also see ==
* Primitive of $x^2 \sqrt {x^2 - a^2}$
* Primitive of $x^2 \sqrt {a^2 - x^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.191$
Category:Primitive of x squared by Root of x squared plus a squared
Category:Primitives involving Root of x squared plus a squared
Primitive of x squared by Root of x squared plus a squared/Also presented as

== Primitive of $x^2 \sqrt {x^2 + a^2}$: Also presented as ==
This result is also seen presented in the form:
:$\ds \int x^2 \sqrt {a^2 + x^2} \rd x = \frac {x \paren {a^2 + 2 x^2} \sqrt {a^2 + x^2} } 8 - \frac {a^4} 8 \arsinh \frac x a + C$
== Proof ==
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $22$.
Category:Primitive of x squared by Root of x squared plus a squared
Primitive of x squared by Root of x squared plus a squared cubed

== Theorem ==
:$\ds \int x^2 \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 + a^2} }^5} 6 - \frac {a^2 x \paren {\sqrt {x^2 + a^2} }^3} {24} - \frac {a^4 x \sqrt {x^2 + a^2} } {16} - \frac {a^6} {16} \map \ln {x + \sqrt {x^2 + a^2} } + C$
== Proof ==
Let:
== Also see ==
* Primitive of $x^2 \paren {\sqrt {x^2 - a^2} }^3$
* Primitive of $x^2 \paren {\sqrt {a^2 - z^2} }^3$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.205$
Category:Primitives involving Root of x squared plus a squared
Primitive of x squared by Sine of a x

== Theorem ==
:$\ds \int x^2 \sin a x \rd x = \frac {2 x \sin a x} {a^2} + \paren {\frac 2 {a^3} - \frac {x^2} a} \cos a x + C$
where $C$ is an arbitrary constant.
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $x^2 \cos a x$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.341$
Category:Primitives involving Sine Function
Primitive of x squared over Cube of Root of a x squared plus b x plus c

== Theorem ==
Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3} = \frac {\paren {2 b^2 - 4 a c} x + 2 b c} {a \paren {4 a c - b^2} \sqrt {a x^2 + b x + c} } + \frac 1 a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$ 
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x^2 + b x + c}$: $14.292$
Category:Primitives involving Root of a x squared plus b x plus c
Primitive of x squared over Root of a squared minus x squared

== Theorem ==
== Proof ==
With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $\dfrac {x^2} {\sqrt {x^2 + a^2} }$
* Primitive of $\dfrac {x^2} {\sqrt {x^2 - a^2} }$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a^2 - x^2}$: $14.239$
* : Front endpapers: A Brief Table of Integrals: $33$.
Category:Primitives involving Root of a squared minus x squared
Primitive of x squared over Root of a squared minus x squared cubed

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\paren {\sqrt {a^2 - x^2} }^3} = \frac x {\sqrt {a^2 - x^2} } - \arcsin \frac x a + C$
== Proof ==
With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $\dfrac {x^2} {\paren {\sqrt {x^2 + a^2} }^3}$
* Primitive of $\dfrac {x^2} {\paren {\sqrt {x^2 - a^2} }^3}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a^2 - x^2}$: $14.253$
Category:Primitives involving Root of a squared minus x squared
Primitive of x squared over Root of a x + b

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\sqrt {a x + b} } = \frac {2 \paren {3 a^2 x^2 - 4 a b x + 8 b^2} \sqrt {a x + b} } {15 a^3}$
== Proof ==
Let:
Thus:
Then:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x + b}$: $14.86$
* : $\S 17$: Tables of Special Indefinite Integrals: $(2)$ Integrals Involving $\sqrt {a x + b}$: $17.2.3.$
Category:Primitives involving Root of a x + b
Primitive of x squared over Root of a x squared plus b x plus c

== Theorem ==
Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {\sqrt {a x^2 + b x + c} } = \frac {2 a x - 3 b} {4 a^2} \sqrt {a x^2 + b x + c} + \frac {3 b^2 - 4 a c} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$
== Proof ==
First:
Then:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x^2 + b x + c}$: $14.282$
Category:Primitives involving Root of a x squared plus b x plus c
Primitive of x squared over Root of x squared minus a squared

== Theorem ==
=== Inverse Hyperbolic Cosine Form ===
=== Logarithm Form ===
== Also see ==
* Primitive of Reciprocal of $\dfrac {x^2} {\sqrt {x^2 + a^2} }$
* Primitive of Reciprocal of $\dfrac {x^2} {\sqrt {a^2 - x^2} }$
Category:Primitives involving Root of x squared minus a squared
Category:Primitive of x squared over Root of x squared minus a squared
Primitive of x squared over Root of x squared minus a squared/Inverse Hyperbolic Cosine Form

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} } = \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \cosh^{-1} \frac x a + C$
for $x > a$.
== Proof ==
With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
Note that because:
:$\cosh^{-1} \dfrac x a$ 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.
== Also see ==
* Primitive of Reciprocal of $\dfrac {x^2} {\sqrt {x^2 + a^2} }$
* Primitive of Reciprocal of $\dfrac {x^2} {\sqrt {a^2 - x^2} }$
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $44$.
Category:Primitive of x squared over Root of x squared minus a squared
Primitive of x squared over Root of x squared minus a squared/Logarithm Form

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} } = \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $\size x > a$.
== Proof ==
With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.211$
* : $\S 17$: Tables of Special Indefinite Integrals: $(9)$ Integrals Involving $\sqrt {x^2 - a^2}$: $17.10.2.$
Category:Primitive of x squared over Root of x squared minus a squared
Primitive of x squared over Root of x squared minus a squared cubed

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \frac {-x} {\sqrt {x^2 - a^2} } + \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $\size x > a$.
== Proof ==
With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $\dfrac {x^2} {\paren {\sqrt {x^2 + a^2} }^3}$
* Primitive of $\dfrac {x^2} {\paren {\sqrt {a^2 - x^2} }^3}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.225$
Category:Primitives involving Root of x squared minus a squared
Primitive of x squared over Root of x squared plus a squared

== Theorem ==
=== Inverse Hyperbolic Sine Form ===
=== Logarithm Form ===
== Also see ==
* Primitive of Reciprocal of $\dfrac {x^2} {\sqrt {x^2 - a^2} }$
* Primitive of Reciprocal of $\dfrac {x^2} {\sqrt {a^2 - x^2} }$
Category:Primitives involving Root of x squared plus a squared
Category:Primitive of x squared over Root of x squared plus a squared
Primitive of x squared over Root of x squared plus a squared/Inverse Hyperbolic Sine Form

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} } = \frac {x \sqrt {x^2 + a^2} } 2 - \frac {a^2} 2 \sinh^{-1} \frac x a + C$
== Proof ==
With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $25$.
Category:Primitive of x squared over Root of x squared plus a squared
Primitive of x squared over Root of x squared plus a squared/Logarithm Form

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} } = \frac {x \sqrt {x^2 + a^2} } 2 - \frac {a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } + C$
== Proof ==
With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.184$
Category:Primitive of x squared over Root of x squared plus a squared
Primitive of x squared over Root of x squared plus a squared cubed

== Theorem ==
:$\ds \int \frac {x^2 \rd x} {\paren {\sqrt {x^2 + a^2} }^3} = \frac {-x} {\sqrt {x^2 + a^2} } + \map \ln {x + \sqrt {x^2 + a^2} } + C$
== Proof ==
With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $\dfrac {x^2} {\paren {\sqrt {x^2 - a^2} }^3}$
* Primitive of $\dfrac {x^2} {\paren {\sqrt {a^2 - x^2} }^3}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.198$
Category:Primitives involving Root of x squared plus a squared
Primitive of x squared over a squared minus x squared/Inverse Hyperbolic Tangent Form

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