What is the integral of (36-x^2)^{5/2} ?

The integral of (36-x^2)^{5/2} is

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integral of (36-x^2)^{5/2}

\int (36 - x^{2})^{\frac{5}{2}} d x

x (36 - x^{2})^{\frac{5}{2}} - (- 233280 (\frac{1}{1296} x (36 - x^{2})^{\frac{3}{2}} + \frac{3}{8} arcsin (\frac{1}{6} x) - \frac{3}{32} sin (4 arcsin (\frac{1}{6} x)) - \frac{1}{279936} x (36 - x^{2})^{\frac{5}{2}} - \frac{5}{31104} x (36 - x^{2})^{\frac{3}{2}} - \frac{5}{32} (2 arcsin (\frac{1}{6} x) + sin (2 arcsin (\frac{1}{6} x))))) + C

Solution steps

\int (36 - x^{2})^{\frac{5}{2}} d x

Apply Integration By Parts

: x (36 - x^{2})^{\frac{5}{2}} - \int - 5 x^{2} (36 - x^{2})^{\frac{3}{2}} d x

= x (36 - x^{2})^{\frac{5}{2}} - \int - 5 x^{2} (36 - x^{2})^{\frac{3}{2}} d x

\int - 5 x^{2} (36 - x^{2})^{\frac{3}{2}} d x = - 233280 (\frac{1}{1296} x (36 - x^{2})^{\frac{3}{2}} + \frac{3}{8} arcsin (\frac{1}{6} x) - \frac{3}{32} sin (4 arcsin (\frac{1}{6} x)) - \frac{1}{279936} x (36 - x^{2})^{\frac{5}{2}} - \frac{5}{31104} x (36 - x^{2})^{\frac{3}{2}} - \frac{5}{32} (2 arcsin (\frac{1}{6} x) + sin (2 arcsin (\frac{1}{6} x))))

= x (36 - x^{2})^{\frac{5}{2}} - (- 233280 (\frac{1}{1296} x (36 - x^{2})^{\frac{3}{2}} + \frac{3}{8} arcsin (\frac{1}{6} x) - \frac{3}{32} sin (4 arcsin (\frac{1}{6} x)) - \frac{1}{279936} x (36 - x^{2})^{\frac{5}{2}} - \frac{5}{31104} x (36 - x^{2})^{\frac{3}{2}} - \frac{5}{32} (2 arcsin (\frac{1}{6} x) + sin (2 arcsin (\frac{1}{6} x)))))

Add a constant to the solution

= x (36 - x^{2})^{\frac{5}{2}} - (- 233280 (\frac{1}{1296} x (36 - x^{2})^{\frac{3}{2}} + \frac{3}{8} arcsin (\frac{1}{6} x) - \frac{3}{32} sin (4 arcsin (\frac{1}{6} x)) - \frac{1}{279936} x (36 - x^{2})^{\frac{5}{2}} - \frac{5}{31104} x (36 - x^{2})^{\frac{3}{2}} - \frac{5}{32} (2 arcsin (\frac{1}{6} x) + sin (2 arcsin (\frac{1}{6} x))))) + C

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