What is the integral of (4x^2)/(sqrt(4-x^2)) ?

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

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integral of (4x^2)/(sqrt(4-x^2))

\int \frac{4 x ^{2}}{4 - x ^{2}} d x

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

Solution steps

\int \frac{4 x ^{2}}{4 - x ^{2}} d x

Take the constant out:

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

= 4 \cdot \int \frac{x ^{2}}{4 - x ^{2}} d x

Apply Trigonometric Substitution

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

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

Take the constant out:

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

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

Rewrite using trig identities

= 4 \cdot 4 \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

= 4 \cdot 4 \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

= 4 \cdot 4 \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)

= 4 \cdot 4 \cdot \frac{1}{2} (u - \frac{1}{2} sin (2 u))

Substitute back

u = arcsin (\frac{1}{2} x)

= 4 \cdot 4 \cdot \frac{1}{2} (arcsin (\frac{1}{2} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} x)))

Simplify

4 \cdot 4 \cdot \frac{1}{2} (arcsin (\frac{1}{2} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} x))) : 8 (arcsin (\frac{1}{2} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} x)))

= 8 (arcsin (\frac{1}{2} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} x)))

Add a constant to the solution

= 8 (arcsin (\frac{1}{2} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} x))) + C

y = \frac{1}{3 x ^{2}} - \frac{5}{2 x}

\int \frac{3}{2} t^{2} d t

f (x) = - 3 x^{2} - 2, at x = 5

\int - 4 x e^{x^{2}} d x

\int x sec^{2} (2 x) d x

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