What is the integral of x^2(9-x^2)^{3/2} ?

The integral of x^2(9-x^2)^{3/2} is

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integral of x^2(9-x^2)^{3/2}

\int x^{2} (9 - x^{2})^{\frac{3}{2}} d x

729 (\frac{1}{81} x (9 - x^{2})^{\frac{3}{2}} + \frac{3}{8} arcsin (\frac{1}{3} x) - \frac{3}{32} sin (4 arcsin (\frac{1}{3} x)) - \frac{1}{4374} x (9 - x^{2})^{\frac{5}{2}} - \frac{5}{1944} x (9 - x^{2})^{\frac{3}{2}} - \frac{5}{32} (2 arcsin (\frac{1}{3} x) + sin (2 arcsin (\frac{1}{3} x)))) + C

Solution steps

\int x^{2} (9 - x^{2})^{\frac{3}{2}} d x

Apply Trigonometric Substitution

: \int 729 cos^{4} (u) sin^{2} (u) d u

= \int 729 cos^{4} (u) sin^{2} (u) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 729 \cdot \int cos^{4} (u) sin^{2} (u) d u

Rewrite using trig identities

= 729 \cdot \int cos^{4} (u) (1 - cos^{2} (u)) d u

Expand

cos^{4} (u) (1 - cos^{2} (u)) : cos^{4} (u) - cos^{6} (u)

= 729 \cdot \int cos^{4} (u) - cos^{6} (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 729 (\int cos^{4} (u) d u - \int cos^{6} (u) d u)

\int cos^{4} (u) d u = cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u))

\int cos^{6} (u) d u = \frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))

= 729 (cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u)) - (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))))

Substitute back

u = arcsin (\frac{1}{3} x)

= 729 (cos^{3} (arcsin (\frac{1}{3} x)) sin (arcsin (\frac{1}{3} x)) + \frac{3}{8} (arcsin (\frac{1}{3} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{3} x))) - (\frac{sin ( arcsin ( \frac{1}{3} x ) ) cos ^{5} ( arcsin ( \frac{1}{3} x ) )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (arcsin (\frac{1}{3} x)) sin (arcsin (\frac{1}{3} x)) + \frac{3}{8} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))))))

Simplify

729 (cos^{3} (arcsin (\frac{1}{3} x)) sin (arcsin (\frac{1}{3} x)) + \frac{3}{8} (arcsin (\frac{1}{3} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{3} x))) - (\frac{sin ( arcsin ( \frac{1}{3} x ) ) cos ^{5} ( arcsin ( \frac{1}{3} x ) )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (arcsin (\frac{1}{3} x)) sin (arcsin (\frac{1}{3} x)) + \frac{3}{8} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x)))))) : 729 (\frac{1}{81} x (9 - x^{2})^{\frac{3}{2}} + \frac{3}{8} arcsin (\frac{1}{3} x) - \frac{3}{32} sin (4 arcsin (\frac{1}{3} x)) - \frac{1}{4374} x (9 - x^{2})^{\frac{5}{2}} - \frac{5}{1944} x (9 - x^{2})^{\frac{3}{2}} - \frac{5}{32} (2 arcsin (\frac{1}{3} x) + sin (2 arcsin (\frac{1}{3} x))))

= 729 (\frac{1}{81} x (9 - x^{2})^{\frac{3}{2}} + \frac{3}{8} arcsin (\frac{1}{3} x) - \frac{3}{32} sin (4 arcsin (\frac{1}{3} x)) - \frac{1}{4374} x (9 - x^{2})^{\frac{5}{2}} - \frac{5}{1944} x (9 - x^{2})^{\frac{3}{2}} - \frac{5}{32} (2 arcsin (\frac{1}{3} x) + sin (2 arcsin (\frac{1}{3} x))))

Add a constant to the solution

= 729 (\frac{1}{81} x (9 - x^{2})^{\frac{3}{2}} + \frac{3}{8} arcsin (\frac{1}{3} x) - \frac{3}{32} sin (4 arcsin (\frac{1}{3} x)) - \frac{1}{4374} x (9 - x^{2})^{\frac{5}{2}} - \frac{5}{1944} x (9 - x^{2})^{\frac{3}{2}} - \frac{5}{32} (2 arcsin (\frac{1}{3} x) + sin (2 arcsin (\frac{1}{3} x)))) + C

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