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

The integral of x^2sqrt(r^2-x^2) is 1/8 r^4(arcsin(x/r)-1/4 sin(4arcsin(x/r)))+C

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integral of x^2sqrt(r^2-x^2)

\int x^{2} r^{2} - x^{2} d x

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

Solution steps

\int x^{2} r^{2} - x^{2} d x

Apply Trigonometric Substitution

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

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

Take the constant out:

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

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

Rewrite using trig identities

= r^{4} \cdot \int \frac{1 - cos ( 4 u )}{8} d u

Take the constant out:

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

= r^{4} \frac{1}{8} \cdot \int 1 - cos (4 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

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

\int 1 d u = u

\int cos (4 u) d u = \frac{1}{4} sin (4 u)

= r^{4} \frac{1}{8} (u - \frac{1}{4} sin (4 u))

Substitute back

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

= r^{4} \frac{1}{8} (arcsin (\frac{1}{r} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{r} x)))

Simplify

r^{4} \frac{1}{8} (arcsin (\frac{1}{r} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{r} x))) : \frac{1}{8} r^{4} (arcsin (\frac{x}{r}) - \frac{1}{4} sin (4 arcsin (\frac{x}{r})))

= \frac{1}{8} r^{4} (arcsin (\frac{x}{r}) - \frac{1}{4} sin (4 arcsin (\frac{x}{r})))

Add a constant to the solution

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

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