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

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

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

\int \frac{x ^{2}}{a ^{2} - x ^{2}} d x

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

Solution steps

\int \frac{x ^{2}}{a ^{2} - x ^{2}} d x

Apply Trigonometric Substitution

= \int a^{2} sin^{2} (u) d u

Take the constant out:

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

= a^{2} \cdot \int sin^{2} (u) d u

Rewrite using trig identities

= a^{2} \cdot \int \frac{1 - cos ( 2 u )}{2} d u

Take the constant out:

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

= a^{2} \frac{1}{2} \cdot \int 1 - cos (2 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

= a^{2} \frac{1}{2} (\int 1 d u - \int cos (2 u) d u)

\int 1 d u = u

\int cos (2 u) d u = \frac{1}{2} sin (2 u)

= a^{2} \frac{1}{2} (u - \frac{1}{2} sin (2 u))

Substitute back

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

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

Simplify

a^{2} \frac{1}{2} (arcsin (\frac{1}{a} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{a} x))) : \frac{1}{2} a^{2} (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a})))

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

Add a constant to the solution

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

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