Integration of Cosine Inverse

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

The integration of cosine inverse is of the form

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

When using integration by parts it must have at least two functions, however this has only one function:  {\cos ^{ - 1}}x. So consider the second function as 1. Now the integration becomes

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

The first function is {\cos ^{ - 1}}x and the second function is 1.

Using the 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} } }

Using the formula above, equation (i) becomes

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

Multiplying and dividing by -2, we have

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

Now we can also use this integration of cosine inverse as a formula.