Primitive of x by Arccosine of x over a/Proof 1
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Theorem
- $\ds \int x \arccos \frac x a \rd x = \paren {\frac {x^2} 2 - \frac {a^2} 4} \arccos \frac x a - \frac {x \sqrt {a^2 - x^2} } 4 + C$
Proof
Let:
\(\ds u\) | \(=\) | \(\ds \arccos \frac x a\) | ||||||||||||
\(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \cos u\) | \(=\) | \(\ds \frac x a\) | Definition of Real Arccosine | |||||||||
\(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \sin u\) | \(=\) | \(\ds \sqrt {1 - \frac {x^2} {a^2} }\) | Sum of Squares of Sine and Cosine |
Then:
\(\ds \int x \arccos \frac x a \rd x\) | \(=\) | \(\ds -a \int u \left({a \cos u}\right) \sin u \rd u\) | 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 | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac {a^2 \sin u \cos u} 4 + \frac {a^2 \arccos \frac x a \paren {\cos^2 u - \sin^2 u} } 4 + C\) | substituting for $u$ | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac {a^2 \sin u \frac x a} 4 + \frac {a^2 \arccos \frac x a \paren {\frac {x^2} {a^2} - \sin^2 u} } 4 + C\) | substituting for $\cos u$ from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac {a^2 \sqrt {1 - \frac {x^2} {a^2} } \frac x a} 4 + \frac {a^2 \arccos \frac x a \paren {\frac {x^2} {a^2} - \paren {1 - \frac {x^2} {a^2} } } } 4 + C\) | substituting for $\sin u$ from $(2)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {x^2} 2 - \frac {a^2} 4} \arccos \frac x a - \frac {x \sqrt {a^2 - x^2} } 4 + C\) | simplifying |
$\blacksquare$