Primitive of x by Arcsine of x over a

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Theorem

$\displaystyle \int x \arcsin \frac x a \ \mathrm d x = \left({\frac {x^2} 2 - \frac {a^2} 4}\right) \arcsin \frac x a + \frac {x \sqrt {a^2 - x^2} } 4 + C$


Proof 1

Let:

\(\displaystyle u\) \(=\) \(\displaystyle \arcsin \frac x a\)
\((1):\quad\) \(\displaystyle \implies \ \ \) \(\displaystyle \sin u\) \(=\) \(\displaystyle \frac x a\) Definition of Arcsine
\((2):\quad\) \(\displaystyle \implies \ \ \) \(\displaystyle \cos u\) \(=\) \(\displaystyle \sqrt {1 - \frac {x^2} {a^2} }\) Sum of Squares of Sine and Cosine


Then:

\(\displaystyle \int x \arcsin \frac x a \ \mathrm d x\) \(=\) \(\displaystyle a \int u \left({a \sin u}\right) \cos u \ \mathrm d u\) Primitive of Function of Arcsine
\(\displaystyle \) \(=\) \(\displaystyle a \int a u \frac {\sin 2 u} 2 \ \mathrm d u\) Double Angle Formula for Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac {a^2} 2 \int u \sin 2 u \ \mathrm d u\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle \frac {a^2} 2 \left({\frac {\sin 2 u} 4 - \frac {u \cos 2 u} 2}\right) + C\) Primitive of $x \sin a x$ where $a = 2$
\(\displaystyle \) \(=\) \(\displaystyle \frac {a^2 \sin 2 u} 8 - \frac {a^2 u \cos 2 u} 4 + C\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac {a^2 \sin u \cos u} 4 - \frac {a^2 u \cos 2 u} 4 + C\) Double Angle Formula for Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac {a^2 \sin u \cos u} 4 - \frac {a^2 u \left({\cos^2 u - \sin^2 u}\right)} 4 + C\) Double Angle Formula for Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac {a^2 \sin u \cos u} 4 - \frac {a^2 \arcsin \frac x a \left({\cos^2 u - \sin^2 u}\right)} 4 + C\) substituting for $u$
\(\displaystyle \) \(=\) \(\displaystyle \frac {a^2 \frac x a \cos u} 4 - \frac {a^2 \arcsin \frac x a \left({\cos^2 u - \frac {x^2} {a^2} }\right)} 4 + C\) substituting for $\sin u$ from $(1)$
\(\displaystyle \) \(=\) \(\displaystyle \frac {a^2 \frac x a \sqrt {1 - \frac {x^2} {a^2} } } 4 - \frac {a^2 \arcsin \frac x a \left({\left({1 - \frac {x^2} {a^2} }\right) - \frac {x^2} {a^2} }\right)} 4 + C\) substituting for $\cos u$ from $(2)$
\(\displaystyle \) \(=\) \(\displaystyle \left({\frac {x^2} 2 - \frac {a^2} 4}\right) \arcsin \frac x a + \frac {x \sqrt {a^2 - x^2} } 4 + C\) simplifying

$\blacksquare$


Proof 2

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:

\(\displaystyle u\) \(=\) \(\displaystyle \arcsin \frac x a\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d u} {\mathrm d x}\) \(=\) \(\displaystyle \frac 1 {\sqrt {a^2 - x^2} }\) Derivative of $\arcsin \dfrac x a$


and let:

\(\displaystyle \frac {\mathrm d v} {\mathrm d x}\) \(=\) \(\displaystyle x\)
\(\displaystyle \implies \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle \frac {x^2} 2\) Primitive of Power


Then:

\(\displaystyle \int x \arcsin \frac x a \ \mathrm d x\) \(=\) \(\displaystyle \frac {x^2} 2 \arcsin \frac x a - \int \frac {x^2} 2 \left({\frac 1 {\sqrt {a^2 - x^2} } }\right) \ \mathrm d x + C\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle \frac {x^2} 2 \arcsin \frac x a - \frac 1 2 \int \frac {x^2 \ \mathrm d x} {\sqrt {a^2 - x^2} } + C\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle \frac {x^2} 2 \arcsin \frac x a - \frac 1 2 \left({\frac {-x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a}\right) + C\) Primitive of $\dfrac {x^2} {\sqrt {a^2 - x^2} }$
\(\displaystyle \) \(=\) \(\displaystyle \left({\frac {x^2} 2 - \frac {a^2} 4}\right) \arcsin \frac x a + \frac {x \sqrt {a^2 - x^2} } 4 + C\) simplifying

$\blacksquare$


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