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

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

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

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

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

Solution steps

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

Apply Trigonometric Substitution

: \int \frac{sin ^{2} ( u )}{2 2} d u

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

Take the constant out:

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

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

Rewrite using trig identities

= \frac{1}{2 2} \cdot \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 2} \cdot \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 2} \cdot \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 2} \cdot \frac{1}{2} (u - \frac{1}{2} sin (2 u))

Substitute back

u = arcsin (2 x)

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

Simplify

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

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (ln (3 x^{2} + 4 y^{2} + 2))

n = 1 \sum \infty x^{n}

\int x^{6} e^{- x} d x

\frac{\partial}{\partial y} (cos (y x^{2}))

- \frac{12}{x ^{6}}

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