Integration of the Square Root of a^2-x^2

In this tutorial we shall derive the integration of the square root of a^2-x^2, and solve this integration with the help of the integration by parts methods.

The integral of \sqrt {{a^2} - {x^2}} is of the form

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

This integral can be written as

I = \int {\sqrt {{a^2} - {x^2}} \cdot 1dx}

Here the first function is \sqrt {{a^2} - {x^2}} and the second function is 1

I = \int {\sqrt {{a^2} - {x^2}} \cdot 1dx} \,\,\,\,{\text{ - - - }}\left( {\text{i}} \right)

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