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

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

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

\int \frac{2}{2 - x ^{2}} d x

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

Solution steps

\int \frac{2}{2 - x ^{2}} d x

\frac{2}{2 - x ^{2}} = 2 \frac{1}{2 - x ^{2}}

= \int 2 \frac{1}{2 - x ^{2}} d x

Take the constant out:

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

= 2 \cdot \int \frac{1}{2 - x ^{2}} d x

Simplify

\frac{1}{2 - x ^{2}} : \frac{1}{2 - x ^{2}}

= 2 \cdot \int \frac{1}{2 - x ^{2}} d x

Take the constant out:

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

= 21 \cdot \int \frac{1}{2 - x ^{2}} d x

Simplify

= 2 \cdot 1 \cdot \int \frac{1}{2 - x ^{2}} d x

Apply Trigonometric Substitution

: \int 1 d u

= 2 \cdot 1 \cdot \int 1 d u

Integral of a constant:

\int a d x = a x

= 2 \cdot 1 \cdot 1 \cdot u

Substitute back

u = arcsin (\frac{1}{2} x)

= 2 \cdot 1 \cdot 1 \cdot arcsin (\frac{1}{2} x)

Simplify

= 2 arcsin (\frac{1}{2} x)

Add a constant to the solution

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

\int \frac{sin ( x )}{cos ^{4} ( x )} d x

\frac{5 x}{( 1 + x ^{2} )}

2 x^{3} + 3 x^{2} - 12 x + 4

\frac{\partial}{\partial x} (x^{2} \cdot y + \frac{x e ^{x}}{y ^{3}})

\int \frac{1}{3 + 2 x ^{2}} d x

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