List of Integrals of Inverse Trigonometric Functions - Arccosine Function Integration Formulas

Arccosine Function Integration Formulas

\int\arccos(a\,x)\,dx= x\arccos(a\,x)- \frac{\sqrt{1-a^2\,x^2}}{a}+C
\int x\arccos(a\,x)\,dx= \frac{x^2\arccos(a\,x)}{2}- \frac{\arccos(a\,x)}{4\,a^2}- \frac{x\sqrt{1-a^2\,x^2}}{4\,a}+C
\int x^2\arccos(a\,x)\,dx= \frac{x^3\arccos(a\,x)}{3}- \frac{\left(a^2\,x^2+2\right)\sqrt{1-a^2\,x^2}}{9\,a^3}+C
\int x^m\arccos(a\,x)\,dx= \frac{x^{m+1}\arccos(a\,x)}{m+1}\,+\, \frac{a}{m+1}\int \frac{x^{m+1}}{\sqrt{1-a^2\,x^2}}\,dx\quad(m\ne-1)


\int\arccos(a\,x)^2\,dx= -2\,x+x\arccos(a\,x)^2- \frac{2\sqrt{1-a^2\,x^2}\arccos(a\,x)}{a}+C
\int\arccos(a\,x)^n\,dx= x\arccos(a\,x)^n\,-\, \frac{n\sqrt{1-a^2\,x^2}\arccos(a\,x)^{n-1}}{a}\,-\, n\,(n-1)\int\arccos(a\,x)^{n-2}\,dx
\int\arccos(a\,x)^n\,dx= \frac{x\arccos(a\,x)^{n+2}}{(n+1)\,(n+2)}\,-\, \frac{\sqrt{1-a^2\,x^2}\arccos(a\,x)^{n+1}}{a\,(n+1)}\,-\, \frac{1}{(n+1)\,(n+2)}\int\arccos(a\,x)^{n+2}\,dx\quad(n\ne-1,-2)

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