What is the integral of sqrt(55-6x-x^2) ?

The integral of sqrt(55-6x-x^2) is 32(arcsin(1/8 (x+3))+1/2 sin(2arcsin(1/8 (x+3))))+C

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integral of sqrt(55-6x-x^2)

\int 55 - 6 x - x^{2} d x

32 (arcsin (\frac{1}{8} (x + 3)) + \frac{1}{2} sin (2 arcsin (\frac{1}{8} (x + 3)))) + C

Solution steps

\int 55 - 6 x - x^{2} d x

Complete the square

55 - 6 x - x^{2} : - (x + 3)^{2} + 64

= \int - (x + 3)^{2} + 64 d x

Apply u-substitution

= \int - u^{2} + 64 d u

Apply Trigonometric Substitution

= \int 64 cos^{2} (v) d v

Take the constant out:

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

= 64 \cdot \int cos^{2} (v) d v

Rewrite using trig identities

= 64 \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

= 64 \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

= 64 \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)

= 64 \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= 64 \cdot \frac{1}{2} (arcsin (\frac{1}{8} (x + 3)) + \frac{1}{2} sin (2 arcsin (\frac{1}{8} (x + 3))))

Simplify

64 \cdot \frac{1}{2} (arcsin (\frac{1}{8} (x + 3)) + \frac{1}{2} sin (2 arcsin (\frac{1}{8} (x + 3)))) : 32 (arcsin (\frac{1}{8} (x + 3)) + \frac{1}{2} sin (2 arcsin (\frac{1}{8} (x + 3))))

= 32 (arcsin (\frac{1}{8} (x + 3)) + \frac{1}{2} sin (2 arcsin (\frac{1}{8} (x + 3))))

Add a constant to the solution

= 32 (arcsin (\frac{1}{8} (x + 3)) + \frac{1}{2} sin (2 arcsin (\frac{1}{8} (x + 3)))) + C

d er i v a t i v e \frac{a x + b}{c x + d}

\frac{d}{d x} (2 x e^{2 x} + 2 x^{2} e^{2 x})

\int a sin (x) d x

\frac{d}{d x} (cos (x^{2} + y^{2}))

a re a y^{2} = 4 x, y = 2 x + 4

What is the integral of sqrt(55-6x-x^2) ?
The integral of sqrt(55-6x-x^2) is 32(arcsin(1/8 (x+3))+1/2 sin(2arcsin(1/8 (x+3))))+C