What is the integral of sqrt((a^2-x^2)^3) ?

The integral of sqrt((a^2-x^2)^3) is x(a^2-x^2)^{3/2}+3/8 a^4(arcsin(x/a)-1/4 sin(4arcsin(x/a)))+C

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integral of sqrt((a^2-x^2)^3)

\int (a^{2} - x^{2})^{3} d x

x (a^{2} - x^{2})^{\frac{3}{2}} + \frac{3}{8} a^{4} (arcsin (\frac{x}{a}) - \frac{1}{4} sin (4 arcsin (\frac{x}{a}))) + C

Solution steps

\int (a^{2} - x^{2})^{3} d x

(a^{2} - x^{2})^{3} = (a^{2} - x^{2})^{\frac{3}{2}},

assuming

(a^{2} - x^{2}) \geq 0

= \int (a^{2} - x^{2})^{\frac{3}{2}} d x

Apply Integration By Parts

: x (a^{2} - x^{2})^{\frac{3}{2}} - \int - 3 x^{2} a^{2} - x^{2} d x

= x (a^{2} - x^{2})^{\frac{3}{2}} - \int - 3 x^{2} a^{2} - x^{2} d x

\int - 3 x^{2} a^{2} - x^{2} d x = - \frac{3}{8} a^{4} (arcsin (\frac{x}{a}) - \frac{1}{4} sin (4 arcsin (\frac{x}{a})))

= x (a^{2} - x^{2})^{\frac{3}{2}} - (- \frac{3}{8} a^{4} (arcsin (\frac{x}{a}) - \frac{1}{4} sin (4 arcsin (\frac{x}{a}))))

Simplify

= x (a^{2} - x^{2})^{\frac{3}{2}} + \frac{3}{8} a^{4} (arcsin (\frac{x}{a}) - \frac{1}{4} sin (4 arcsin (\frac{x}{a})))

Add a constant to the solution

= x (a^{2} - x^{2})^{\frac{3}{2}} + \frac{3}{8} a^{4} (arcsin (\frac{x}{a}) - \frac{1}{4} sin (4 arcsin (\frac{x}{a}))) + C

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What is the integral of sqrt((a^2-x^2)^3) ?
The integral of sqrt((a^2-x^2)^3) is x(a^2-x^2)^{3/2}+3/8 a^4(arcsin(x/a)-1/4 sin(4arcsin(x/a)))+C