Primitive of x by Arcsine of x over a

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Theorem

$\ds \int x \arcsin \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arcsin \frac x a + \frac {x \sqrt {a^2 - x^2} } 4 + C$


Proof 1

Let:

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


Then:

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

$\blacksquare$


Proof 2

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

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


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds x\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {x^2} 2\) Primitive of Power


Then:

\(\ds \int x \arcsin \frac x a \rd x\) \(=\) \(\ds \frac {x^2} 2 \arcsin \frac x a - \int \frac {x^2} 2 \paren {\frac 1 {\sqrt {a^2 - x^2} } } \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac {x^2} 2 \arcsin \frac x a - \frac 1 2 \int \frac {x^2 \rd x} {\sqrt {a^2 - x^2} } + C\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds \frac {x^2} 2 \arcsin \frac x a - \frac 1 2 \paren {\frac {-x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a} + C\) Primitive of $\dfrac {x^2} {\sqrt {a^2 - x^2} }$
\(\ds \) \(=\) \(\ds \paren {\frac {x^2} 2 - \frac {a^2} 4} \arcsin \frac x a + \frac {x \sqrt {a^2 - x^2} } 4 + C\) simplifying

$\blacksquare$


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