Integration of Sine Inverse

In this tutorial we shall explain integration of sine inverse function {\sin  ^{ - 1}}x it is an important integral function, but it has no direct method to find we shall find the integration of sine inverse by using integration by parts method.
The integration of sine inverse is of the form

I =  \int {{{\sin }^{ - 1}}xdx}


By using integration by parts, it must have at least two functions, it in this function it has only one function that is {\sin ^{ - 1}}x, now consider second function as 1. Now the integration becomes

I =  \int {{{\sin }^{ - 1}}x \cdot 1dx} \,\,\,\,{\text{ -  -  -  }}\left( {\text{i}} \right)


Take first function is {\sin  ^{ - 1}}x and second function will be 1
Using formula for integration by parts, we have

\int  {\left[ {f\left( x \right)g\left( x \right)} \right]dx = f\left( x \right)\int  {g\left( x \right)dx - \int {\left[ {\frac{d}{{dx}}f\left( x \right)\int  {g\left( x \right)dx} } \right]dx} } }


Equation (i) becomes using above formula, we have

\begin{gathered} I = {\sin ^{ - 1}}x\int {1dx - \int {\left[  {\frac{d}{{dx}}{{\sin }^{ - 1}}x\int {1dx} } \right]dx} } \\ \Rightarrow I = x{\sin ^{ - 1}}x - \int  {\left[ {\frac{1}{{\sqrt {1 - {x^2}} }}x} \right]dx} \\ \Rightarrow I = x{\sin ^{ - 1}}x - \int  {{{\left( {1 - {x^2}} \right)}^{ - \frac{1}{2}}}xdx} \\ \end{gathered}


Multiplying and dividing by -2, we have

I =  x{\sin ^{ - 1}}x + \frac{1}{2}\int {{{\left( {1 - {x^2}} \right)}^{ -  \frac{1}{2}}}\left( { - 2x} \right)dx}


Using formula

\int  {{{\left[ {f\left( x \right)} \right]}^n}f'\left( x \right)dx = \frac{{{{\left[  {f\left( x \right)} \right]}^{n + 1}}}}{{n + 1}}}  + c


\begin{gathered} I = x{\sin ^{ - 1}}x + \frac{{{{\left( {1 -  {x^2}} \right)}^{ - \frac{1}{2} + 1}}}}{{ - \frac{1}{2} + 1}} + c \\ \Rightarrow I = x{\sin ^{ - 1}}x +  \frac{{{{\left( {1 - {x^2}} \right)}^{\frac{1}{2}}}}}{{\frac{1}{2}}} + c \\ \Rightarrow I = x{\sin ^{ - 1}}x + 2\sqrt {1  - {x^2}}  + c \\ \Rightarrow \int {{{\sin }^{ - 1}}xdx}  = x{\sin ^{ - 1}}x + 2\sqrt {1 - {x^2}}  + c \\ \end{gathered}


Now in further study in integration we can use this integration of sine inverse as a formula.