Inverse Trigonometric Functions - Indefinite Integrals of Inverse Trigonometric Functions

Indefinite Integrals of Inverse Trigonometric Functions

For real and complex values of x:


\begin{align}
\int \arcsin x\,dx &{}= x\,\arcsin x + \sqrt{1-x^2} + C\\
\int \arccos x\,dx &{}= x\,\arccos x - \sqrt{1-x^2} + C\\
\int \arctan x\,dx &{}= x\,\arctan x - \frac{1}{2}\ln\left(1+x^2\right) + C\\
\int \arccot x\,dx &{}= x\,\arccot x + \frac{1}{2}\ln\left(1+x^2\right) + C\\
\int \arcsec x\,dx &{}= x\,\arcsec x - \ln\left + C\\
\int \arccsc x\,dx &{}= x\,\arccsc x + \ln\left + C
\end{align}

For real x ≥ 1:


\begin{align}
\int \arcsec x\,dx &{}= x\,\arcsec x - \ln\left(x+\sqrt{x^2-1}\right) + C\\
\int \arccsc x\,dx &{}= x\,\arccsc x + \ln\left(x+\sqrt{x^2-1}\right) + C
\end{align}

All of these can be derived using integration by parts and the simple derivative forms shown above.

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