What is the integral of (x^2)/(sqrt(4-9x^6)) ?

The integral of (x^2)/(sqrt(4-9x^6)) is 1/9 arcsin(3/2 x^3)+C

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integral of (x^2)/(sqrt(4-9x^6))

\int \frac{x ^{2}}{4 - 9 x ^{6}} d x

\frac{1}{9} arcsin (\frac{3}{2} x^{3}) + C

Solution steps

\int \frac{x ^{2}}{4 - 9 x ^{6}} d x

Apply u-substitution

: \int \frac{1}{3 4 - 9 u ^{2}} d u

= \int \frac{1}{3 4 - 9 u ^{2}} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{4 - 9 u ^{2}} d u

Apply Trigonometric Substitution

: \int \frac{1}{3} d v

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

Integral of a constant:

\int a d x = a x

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

Substitute back

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

Simplify

\frac{1}{3} \cdot \frac{1}{3} \cdot 1 \cdot arcsin (\frac{3}{2} x^{3}) : \frac{1}{9} arcsin (\frac{3}{2} x^{3})

= \frac{1}{9} arcsin (\frac{3}{2} x^{3})

Add a constant to the solution

= \frac{1}{9} arcsin (\frac{3}{2} x^{3}) + C

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What is the integral of (x^2)/(sqrt(4-9x^6)) ?
The integral of (x^2)/(sqrt(4-9x^6)) is 1/9 arcsin(3/2 x^3)+C