What is the integral of sqrt(100-4x^2) ?

The integral of sqrt(100-4x^2) is 25(arcsin(x/5)+1/2 sin(2arcsin(x/5)))+C

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integral of sqrt(100-4x^2)

\int 100 - 4 x^{2} d x

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

Solution steps

\int 100 - 4 x^{2} d x

Apply Trigonometric Substitution

= \int 50 cos^{2} (u) d u

Take the constant out:

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

= 50 \cdot \int cos^{2} (u) d u

Rewrite using trig identities

= 50 \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

= 50 \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

= 50 \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)

= 50 \cdot \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Substitute back

u = arcsin (\frac{2}{10} x)

= 50 \cdot \frac{1}{2} (arcsin (\frac{2}{10} x) + \frac{1}{2} sin (2 arcsin (\frac{2}{10} x)))

Simplify

50 \cdot \frac{1}{2} (arcsin (\frac{2}{10} x) + \frac{1}{2} sin (2 arcsin (\frac{2}{10} x))) : 25 (arcsin (\frac{x}{5}) + \frac{1}{2} sin (2 arcsin (\frac{x}{5})))

= 25 (arcsin (\frac{x}{5}) + \frac{1}{2} sin (2 arcsin (\frac{x}{5})))

Add a constant to the solution

= 25 (arcsin (\frac{x}{5}) + \frac{1}{2} sin (2 arcsin (\frac{x}{5}))) + C

What is the integral of sqrt(100-4x^2) ?
The integral of sqrt(100-4x^2) is 25(arcsin(x/5)+1/2 sin(2arcsin(x/5)))+C