What is the integral of sqrt(1-(x-2)^2) ?

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

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integral of sqrt(1-(x-2)^2)

\int 1 - (x - 2)^{2} d x

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

Solution steps

\int 1 - (x - 2)^{2} d x

Expand

1 - (x - 2)^{2} : - x^{2} + 4 x - 3

= \int - x^{2} + 4 x - 3 d x

Complete the square

- x^{2} + 4 x - 3 : - (x - 2)^{2} + 1

= \int - (x - 2)^{2} + 1 d x

Apply u-substitution

= \int - u^{2} + 1 d u

Apply Trigonometric Substitution

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

Rewrite using trig identities

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

Substitute back

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

Add a constant to the solution

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

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