Primitive of Arcsine of x over a

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Theorem

$\displaystyle \int \arcsin \frac x a \rd x = x \arcsin \frac x a + \sqrt {a^2 - x^2} + C$


Proof 1

Let:

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


Then:

\(\displaystyle \int \arcsin \frac x a \rd x\) \(=\) \(\displaystyle a \int u \cos u \rd u\) Primitive of Function of Arcsine
\(\displaystyle \) \(=\) \(\displaystyle a \paren {\cos u + u \sin u} + C\) Primitive of $x \cos a x$
\(\displaystyle \) \(=\) \(\displaystyle a \paren {\cos u + u \frac x a} + C\) Substitution for $\sin u$ from $\paren 1$
\(\displaystyle \) \(=\) \(\displaystyle a \paren {\sqrt {1 - \frac {x^2} {a^2} } + u \frac x a} + C\) Substitution for $\cos u$ from $\paren 2$
\(\displaystyle \) \(=\) \(\displaystyle a \paren {\sqrt {1 - \frac {x^2} {a^2} } + \frac x a \arcsin \frac x a} + C\) Substitution for $u$
\(\displaystyle \) \(=\) \(\displaystyle x \arcsin \frac x a + \sqrt {a^2 - x^2} + C\) simplifying

$\blacksquare$


Proof 2

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:

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


and let:

\(\displaystyle \frac {\d v} {\d x}\) \(=\) \(\displaystyle 1\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle x\) Primitive of Constant


Then:

\(\displaystyle \int \arcsin \frac x a \rd x\) \(=\) \(\displaystyle x \arcsin \frac x a - \int x \paren {\frac 1 {\sqrt {a^2 - x^2} } } \rd x + C\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle x \arcsin \frac x a - \int \frac {x \rd x} {\sqrt {a^2 - x^2} } + C\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle x \arcsin \frac x a - \paren {-\sqrt {a^2 - x^2} } + C\) Primitive of $\dfrac x {\sqrt {a^2 - x^2} }$
\(\displaystyle \) \(=\) \(\displaystyle x \arcsin \frac x a + \sqrt {a^2 - x^2} + C\) simplifying

$\blacksquare$


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