What is the integral of sqrt(27-6x-x^2) ?

The integral of sqrt(27-6x-x^2) is 18(arcsin(1/6 (x+3))+1/2 sin(2arcsin(1/6 (x+3))))+C

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integral of sqrt(27-6x-x^2)

\int 27 - 6 x - x^{2} d x

18 (arcsin (\frac{1}{6} (x + 3)) + \frac{1}{2} sin (2 arcsin (\frac{1}{6} (x + 3)))) + C

Solution steps

\int 27 - 6 x - x^{2} d x

Complete the square

27 - 6 x - x^{2} : - (x + 3)^{2} + 36

= \int - (x + 3)^{2} + 36 d x

Apply u-substitution

: \int - u^{2} + 36 d u

= \int - u^{2} + 36 d u

Apply Trigonometric Substitution

: \int 36 cos^{2} (v) d v

= \int 36 cos^{2} (v) d v

Take the constant out:

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

= 36 \cdot \int cos^{2} (v) d v

Rewrite using trig identities

= 36 \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

= 36 \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

= 36 \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)

= 36 \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= 36 \cdot \frac{1}{2} (arcsin (\frac{1}{6} (x + 3)) + \frac{1}{2} sin (2 arcsin (\frac{1}{6} (x + 3))))

Simplify

36 \cdot \frac{1}{2} (arcsin (\frac{1}{6} (x + 3)) + \frac{1}{2} sin (2 arcsin (\frac{1}{6} (x + 3)))) : 18 (arcsin (\frac{1}{6} (x + 3)) + \frac{1}{2} sin (2 arcsin (\frac{1}{6} (x + 3))))

= 18 (arcsin (\frac{1}{6} (x + 3)) + \frac{1}{2} sin (2 arcsin (\frac{1}{6} (x + 3))))

Add a constant to the solution

= 18 (arcsin (\frac{1}{6} (x + 3)) + \frac{1}{2} sin (2 arcsin (\frac{1}{6} (x + 3)))) + C

\frac{d}{d x} (e^{3 x} \cdot cos (2 x))

\frac{d y}{d x} + y^{5} x + 9 y = 0

\int \frac{7}{x ^{5}} d x

x^{3} + \frac{1}{12 x}

x \to 5 lim (\frac{1}{x ^{2} - 25})

What is the integral of sqrt(27-6x-x^2) ?
The integral of sqrt(27-6x-x^2) is 18(arcsin(1/6 (x+3))+1/2 sin(2arcsin(1/6 (x+3))))+C