What is the integral of (12x^3)/(sqrt(2-x^4)) ?

The integral of (12x^3)/(sqrt(2-x^4)) is -6sqrt(2-x^4)+C

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integral of (12x^3)/(sqrt(2-x^4))

\int \frac{12 x ^{3}}{2 - x ^{4}} d x

- 6 2 - x^{4} + C

Solution steps

\int \frac{12 x ^{3}}{2 - x ^{4}} d x

Take the constant out:

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

= 12 \cdot \int \frac{x ^{3}}{2 - x ^{4}} d x

Apply u-substitution

: \int - \frac{1}{4 u} d u

= 12 \cdot \int - \frac{1}{4 u} d u

Take the constant out:

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

= 12 (- \frac{1}{4} \cdot \int \frac{1}{u} d u)

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= 12 (- \frac{1}{4} \cdot 2 u^{\frac{1}{2}})

Substitute back

u = 2 - x^{4}

= 12 (- \frac{1}{4} \cdot 2 (2 - x^{4})^{\frac{1}{2}})

Simplify

12 (- \frac{1}{4} \cdot 2 (2 - x^{4})^{\frac{1}{2}}) : - 6 2 - x^{4}

= - 6 2 - x^{4}

Add a constant to the solution

= - 6 2 - x^{4} + C

f (x) = 5 x^{4} + \frac{1}{( 2 x ^{2} )}

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x \to 4 lim (3 x - 7)

sin (x) + csc (x)

\int \frac{x}{3 x + 6} d x

What is the integral of (12x^3)/(sqrt(2-x^4)) ?
The integral of (12x^3)/(sqrt(2-x^4)) is -6sqrt(2-x^4)+C