Primitive of Arccosine of x over a over x
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Theorem
- $\ds \int \dfrac 1 x \arccos \frac x a \rd x = \frac \pi 2 \ln \size x - \int \frac {\arcsin \frac x a \rd x} x + C$
Proof
\(\ds \int \dfrac 1 x \arccos \frac x a \rd x\) | \(=\) | \(\ds \int \dfrac 1 x \paren {\frac \pi 2 - \arcsin \frac x a} \rd x\) | Sum of Arcsine and Arccosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 2 \int \frac {\d x} x - \int \dfrac 1 x \arcsin \frac x a \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 2 \ln \size x - \int \dfrac 1 x \arcsin \frac x a \rd x + C\) | Primitive of Reciprocal |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.480$