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Antiderivative of Inverse Trigonometric Functions

The formulas to compute antiderivatives of inverse trigonometric functiosn are deduced.

Antiderivative of the Arcsine function

Define y = \arcsin v. Then, v = \sin y. From this, it follows that dv = \cos y \cdot dy.

In order to calculate the antiderivative for the arcsine function, change the expression in terms of the variable v to its equivalent in terms of y in the integrand as follows:

    \begin{equation*} 	\int \arcsin v \cdot dv = \int y\cdot \cos y\cdot dy \end{equation*}

Now, define u = y, so that du = dy and also let dw = \cos y\cdot dy, and from this, w = \sin y. With this definitions at hand
apply the technique of integration by parts to obtain:

    \begin{equation*} 	\int y\cdot \cos y\cdot dy = y\,\sin y - \int \sin y\cdot dy	 		= y\,\sin y - \cos y + \hat{C} \end{equation*}

Since by definition, v = \sin y, and also, \sin^2 y + \cos^2 y = 1, it follows that \cos^2y = 1 - v^2, therefore,

    \begin{equation*} 	\int \arcsin v \cdot dv = v\cdot \arcsin v + \sqrt{1 - v^2} + C \end{equation*}

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