What is the integral of sqrt(30x-9x^2-21) ?

The integral of sqrt(30x-9x^2-21) is 1/3 (2arcsin(3/2 x-5/2)+sin(2arcsin(3/2 x-5/2)))+C

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integral of sqrt(30x-9x^2-21)

\int 30 x - 9 x^{2} - 21 d x

\frac{1}{3} (2 arcsin (\frac{3}{2} x - \frac{5}{2}) + sin (2 arcsin (\frac{3}{2} x - \frac{5}{2}))) + C

Solution steps

\int 30 x - 9 x^{2} - 21 d x

Complete the square

30 x - 9 x^{2} - 21 : - 9 (x - \frac{5}{3})^{2} + 4

= \int - 9 (x - \frac{5}{3})^{2} + 4 d x

Apply u-substitution

: \int - 9 u^{2} + 4 d u

= \int - 9 u^{2} + 4 d u

Apply Trigonometric Substitution

: \int \frac{4}{3} cos^{2} (v) d v

= \int \frac{4}{3} cos^{2} (v) d v

Take the constant out:

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

= \frac{4}{3} \cdot \int cos^{2} (v) d v

Rewrite using trig identities

= \frac{4}{3} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

Take the constant out:

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

= \frac{4}{3} \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{4}{3} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= \frac{4}{3} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= \frac{4}{3} \cdot \frac{1}{2} (arcsin (\frac{3}{2} (x - \frac{5}{3})) + \frac{1}{2} sin (2 arcsin (\frac{3}{2} (x - \frac{5}{3}))))

Simplify

\frac{4}{3} \cdot \frac{1}{2} (arcsin (\frac{3}{2} (x - \frac{5}{3})) + \frac{1}{2} sin (2 arcsin (\frac{3}{2} (x - \frac{5}{3})))) : \frac{1}{3} (2 arcsin (\frac{3}{2} x - \frac{5}{2}) + sin (2 arcsin (\frac{3}{2} x - \frac{5}{2})))

= \frac{1}{3} (2 arcsin (\frac{3}{2} x - \frac{5}{2}) + sin (2 arcsin (\frac{3}{2} x - \frac{5}{2})))

Add a constant to the solution

= \frac{1}{3} (2 arcsin (\frac{3}{2} x - \frac{5}{2}) + sin (2 arcsin (\frac{3}{2} x - \frac{5}{2}))) + C

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What is the integral of sqrt(30x-9x^2-21) ?
The integral of sqrt(30x-9x^2-21) is 1/3 (2arcsin(3/2 x-5/2)+sin(2arcsin(3/2 x-5/2)))+C