What is the integral of sqrt(25-5x^2) ?

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

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integral of sqrt(25-5x^2)

\int 25 - 5 x^{2} d x

\frac{5 5}{4} (2 arcsin (\frac{x}{5}) + sin (2 arcsin (\frac{x}{5}))) + C

Solution steps

\int 25 - 5 x^{2} d x

Apply Trigonometric Substitution

: \int 55 cos^{2} (u) d u

= \int 55 cos^{2} (u) d u

Take the constant out:

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

= 55 \cdot \int cos^{2} (u) d u

Rewrite using trig identities

= 55 \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

= 55 \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

= 55 \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)

= 55 \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Substitute back

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

= 55 \frac{1}{2} (arcsin (\frac{5}{5} x) + \frac{1}{2} sin (2 arcsin (\frac{5}{5} x)))

Simplify

55 \frac{1}{2} (arcsin (\frac{5}{5} x) + \frac{1}{2} sin (2 arcsin (\frac{5}{5} x))) : \frac{5 5}{4} (2 arcsin (\frac{x}{5}) + sin (2 arcsin (\frac{x}{5})))

= \frac{5 5}{4} (2 arcsin (\frac{x}{5}) + sin (2 arcsin (\frac{x}{5})))

Add a constant to the solution

= \frac{5 5}{4} (2 arcsin (\frac{x}{5}) + sin (2 arcsin (\frac{x}{5}))) + C

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