What is the integral of (x^2)/(sqrt(5-x^2)) ?

The integral of (x^2)/(sqrt(5-x^2)) is 5/2 (arcsin(1/(sqrt(5))x)-1/2 sin(2arcsin(1/(sqrt(5))x)))+C

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integral of (x^2)/(sqrt(5-x^2))

\int \frac{x ^{2}}{5 - x ^{2}} d x

\frac{5}{2} (arcsin (\frac{1}{5} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} x))) + C

Solution steps

\int \frac{x ^{2}}{5 - x ^{2}} d x

Apply Trigonometric Substitution

: \int 5 sin^{2} (u) d u

= \int 5 sin^{2} (u) d u

Take the constant out:

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

= 5 \cdot \int sin^{2} (u) d u

Rewrite using trig identities

= 5 \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

= 5 \cdot \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

= 5 \cdot \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)

= 5 \cdot \frac{1}{2} (u - \frac{1}{2} sin (2 u))

Substitute back

u = arcsin (\frac{1}{5} x)

= 5 \cdot \frac{1}{2} (arcsin (\frac{1}{5} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} x)))

Simplify

5 \cdot \frac{1}{2} (arcsin (\frac{1}{5} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} x))) : \frac{5}{2} (arcsin (\frac{1}{5} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} x)))

= \frac{5}{2} (arcsin (\frac{1}{5} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} x)))

Add a constant to the solution

= \frac{5}{2} (arcsin (\frac{1}{5} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} x))) + C

x \to + π lim (\frac{5 x}{3})

y = \frac{8 x ^{2} + 8 x + 4}{x}

\int 5^{x} d x

\frac{\partial}{\partial x} (arctan (cos (x)))

\frac{\partial}{\partial y} (3 (x - y) e^{3 y + 2 x^{2}})

What is the integral of (x^2)/(sqrt(5-x^2)) ?
The integral of (x^2)/(sqrt(5-x^2)) is 5/2 (arcsin(1/(sqrt(5))x)-1/2 sin(2arcsin(1/(sqrt(5))x)))+C