What is the integral of x/(sqrt(4-3x^4)) ?

The integral of x/(sqrt(4-3x^4)) is 1/(2sqrt(3))arcsin((sqrt(3))/2 x^2)+C

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integral of x/(sqrt(4-3x^4))

\int \frac{x}{4 - 3 x ^{4}} d x

\frac{1}{2 3} arcsin (\frac{3}{2} x^{2}) + C

Solution steps

\int \frac{x}{4 - 3 x ^{4}} d x

Apply u-substitution

= \int \frac{1}{2 4 - 3 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 \frac{1}{4 - 3 u ^{2}} d u

Apply Trigonometric Substitution

= \frac{1}{2} \cdot \int \frac{1}{3} d v

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{3} \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= \frac{1}{2} \cdot \frac{1}{3} \cdot 1 \cdot v

Substitute back

= \frac{1}{2} \cdot \frac{1}{3} \cdot 1 \cdot arcsin (\frac{3}{2} x^{2})

Simplify

\frac{1}{2} \cdot \frac{1}{3} \cdot 1 \cdot arcsin (\frac{3}{2} x^{2}) : \frac{1}{2 3} arcsin (\frac{3}{2} x^{2})

= \frac{1}{2 3} arcsin (\frac{3}{2} x^{2})

Add a constant to the solution

= \frac{1}{2 3} arcsin (\frac{3}{2} x^{2}) + C

What is the integral of x/(sqrt(4-3x^4)) ?
The integral of x/(sqrt(4-3x^4)) is 1/(2sqrt(3))arcsin((sqrt(3))/2 x^2)+C