What is the integral of sqrt(12+4x-x^2) ?

The integral of sqrt(12+4x-x^2) is 8(arcsin(1/4 (x-2))+1/2 sin(2arcsin(1/4 (x-2))))+C

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integral of sqrt(12+4x-x^2)

\int 12 + 4 x - x^{2} d x

8 (arcsin (\frac{1}{4} (x - 2)) + \frac{1}{2} sin (2 arcsin (\frac{1}{4} (x - 2)))) + C

Solution steps

\int 12 + 4 x - x^{2} d x

Complete the square

12 + 4 x - x^{2} : - (x - 2)^{2} + 16

= \int - (x - 2)^{2} + 16 d x

Apply u-substitution

: \int - u^{2} + 16 d u

= \int - u^{2} + 16 d u

Apply Trigonometric Substitution

: \int 16 cos^{2} (v) d v

= \int 16 cos^{2} (v) d v

Take the constant out:

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

= 16 \cdot \int cos^{2} (v) d v

Rewrite using trig identities

= 16 \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

= 16 \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

= 16 \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)

= 16 \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= 16 \cdot \frac{1}{2} (arcsin (\frac{1}{4} (x - 2)) + \frac{1}{2} sin (2 arcsin (\frac{1}{4} (x - 2))))

Simplify

16 \cdot \frac{1}{2} (arcsin (\frac{1}{4} (x - 2)) + \frac{1}{2} sin (2 arcsin (\frac{1}{4} (x - 2)))) : 8 (arcsin (\frac{1}{4} (x - 2)) + \frac{1}{2} sin (2 arcsin (\frac{1}{4} (x - 2))))

= 8 (arcsin (\frac{1}{4} (x - 2)) + \frac{1}{2} sin (2 arcsin (\frac{1}{4} (x - 2))))

Add a constant to the solution

= 8 (arcsin (\frac{1}{4} (x - 2)) + \frac{1}{2} sin (2 arcsin (\frac{1}{4} (x - 2)))) + C

f (x) = 2 x^{2} + 3, (- 1, 2)

x \to 2 lim (3 x^{2} - 1)

x \to 5 + lim (\frac{x + 1}{x - 5})

f (x) = 8 e^{x} + x, at x = 8

x \to \infty lim (- 1 + e^{(x + 1)})

What is the integral of sqrt(12+4x-x^2) ?
The integral of sqrt(12+4x-x^2) is 8(arcsin(1/4 (x-2))+1/2 sin(2arcsin(1/4 (x-2))))+C