What is the integral of (1-y^2)^{1/2} ?

The integral of (1-y^2)^{1/2} is 1/2 (arcsin(y)+1/2 sin(2arcsin(y)))+C

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integral of (1-y^2)^{1/2}

\int (1 - y^{2})^{\frac{1}{2}} d y

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

Solution steps

\int (1 - y^{2})^{\frac{1}{2}} d y

Apply Trigonometric Substitution

: \int cos^{2} (u) d u

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

Rewrite using trig identities

= \int \frac{1 + cos ( 2 u )}{2} d u

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 u) d u

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 u + \int cos (2 u) d u)

\int 1 d u = u

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

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

Substitute back

u = arcsin (y)

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

Add a constant to the solution

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

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What is the integral of (1-y^2)^{1/2} ?
The integral of (1-y^2)^{1/2} is 1/2 (arcsin(y)+1/2 sin(2arcsin(y)))+C