What is the integral of (24x^2)/(sqrt(36-x^2)) ?

The integral of (24x^2)/(sqrt(36-x^2)) is 432(arcsin(1/6 x)-1/2 sin(2arcsin(1/6 x)))+C

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integral of (24x^2)/(sqrt(36-x^2))

\int \frac{24 x ^{2}}{36 - x ^{2}} d x

432 (arcsin (\frac{1}{6} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{6} x))) + C

Solution steps

\int \frac{24 x ^{2}}{36 - x ^{2}} d x

Take the constant out:

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

= 24 \cdot \int \frac{x ^{2}}{36 - x ^{2}} d x

Apply Trigonometric Substitution

: \int 36 sin^{2} (u) d u

= 24 \cdot \int 36 sin^{2} (u) d u

Take the constant out:

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

= 24 \cdot 36 \cdot \int sin^{2} (u) d u

Rewrite using trig identities

= 24 \cdot 36 \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

= 24 \cdot 36 \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

= 24 \cdot 36 \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)

= 24 \cdot 36 \cdot \frac{1}{2} (u - \frac{1}{2} sin (2 u))

Substitute back

u = arcsin (\frac{1}{6} x)

= 24 \cdot 36 \cdot \frac{1}{2} (arcsin (\frac{1}{6} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{6} x)))

Simplify

24 \cdot 36 \cdot \frac{1}{2} (arcsin (\frac{1}{6} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{6} x))) : 432 (arcsin (\frac{1}{6} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{6} x)))

= 432 (arcsin (\frac{1}{6} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{6} x)))

Add a constant to the solution

= 432 (arcsin (\frac{1}{6} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{6} x))) + C

x \to 1 lim (6 x + 4)

\frac{( e ^{\frac{s}{2}} )}{s ^{2}}

x = y^{2} - 7, x = 1 - y^{2}, - 2 \leq y \leq 2

\frac{d y}{d t} t + y = 2 t

\frac{d}{d x} (\frac{x ^{2} - 4}{x ^{2} - 1})

What is the integral of (24x^2)/(sqrt(36-x^2)) ?
The integral of (24x^2)/(sqrt(36-x^2)) is 432(arcsin(1/6 x)-1/2 sin(2arcsin(1/6 x)))+C