What is the integral of (a^2-x^2)^{1/2} ?

The integral of (a^2-x^2)^{1/2} is x(a^2-x^2)^{1/2}+1/2 a^2(arcsin(x/a)-1/2 sin(2arcsin(x/a)))+C

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integral of (a^2-x^2)^{1/2}

\int (a^{2} - x^{2})^{\frac{1}{2}} d x

x (a^{2} - x^{2})^{\frac{1}{2}} + \frac{1}{2} a^{2} (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a}))) + C

Solution steps

\int (a^{2} - x^{2})^{\frac{1}{2}} d x

Apply Integration By Parts

= x (a^{2} - x^{2})^{\frac{1}{2}} - \int - \frac{x ^{2}}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} d x

\int - \frac{x ^{2}}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} d x = - \frac{1}{2} a^{2} (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a})))

= x (a^{2} - x^{2})^{\frac{1}{2}} - (- \frac{1}{2} a^{2} (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a}))))

Simplify

= x (a^{2} - x^{2})^{\frac{1}{2}} + \frac{1}{2} a^{2} (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a})))

Add a constant to the solution

= x (a^{2} - x^{2})^{\frac{1}{2}} + \frac{1}{2} a^{2} (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a}))) + C

What is the integral of (a^2-x^2)^{1/2} ?
The integral of (a^2-x^2)^{1/2} is x(a^2-x^2)^{1/2}+1/2 a^2(arcsin(x/a)-1/2 sin(2arcsin(x/a)))+C