What is the integral of (6-x^2)^{3/2} ?

The integral of (6-x^2)^{3/2} is

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integral of (6-x^2)^{3/2}

\int (6 - x^{2})^{\frac{3}{2}} d x

x (6 - x^{2})^{\frac{3}{2}} + 3 (\frac{1}{4} x^{3} (- x^{2} + 6)^{\frac{1}{2}} + \frac{9}{2} arcsin (\frac{1}{6 ^{\frac{1}{2}}} x) - \frac{9}{4} sin (2 arcsin (\frac{1}{6 ^{\frac{1}{2}}} x))) + C

Solution steps

\int (6 - x^{2})^{\frac{3}{2}} d x

Apply Integration By Parts

= x (6 - x^{2})^{\frac{3}{2}} - \int - 3 x^{2} (6 - x^{2})^{\frac{1}{2}} d x

\int - 3 x^{2} (6 - x^{2})^{\frac{1}{2}} d x = - 3 (\frac{1}{4} x^{3} (- x^{2} + 6)^{\frac{1}{2}} + \frac{9}{2} arcsin (\frac{1}{6 ^{\frac{1}{2}}} x) - \frac{9}{4} sin (2 arcsin (\frac{1}{6 ^{\frac{1}{2}}} x)))

= x (6 - x^{2})^{\frac{3}{2}} - (- 3 (\frac{1}{4} x^{3} (- x^{2} + 6)^{\frac{1}{2}} + \frac{9}{2} arcsin (\frac{1}{6 ^{\frac{1}{2}}} x) - \frac{9}{4} sin (2 arcsin (\frac{1}{6 ^{\frac{1}{2}}} x))))

Simplify

= x (6 - x^{2})^{\frac{3}{2}} + 3 (\frac{1}{4} x^{3} (- x^{2} + 6)^{\frac{1}{2}} + \frac{9}{2} arcsin (\frac{1}{6 ^{\frac{1}{2}}} x) - \frac{9}{4} sin (2 arcsin (\frac{1}{6 ^{\frac{1}{2}}} x)))

Add a constant to the solution

= x (6 - x^{2})^{\frac{3}{2}} + 3 (\frac{1}{4} x^{3} (- x^{2} + 6)^{\frac{1}{2}} + \frac{9}{2} arcsin (\frac{1}{6 ^{\frac{1}{2}}} x) - \frac{9}{4} sin (2 arcsin (\frac{1}{6 ^{\frac{1}{2}}} x))) + C