What is the integral of (3x)/((x^2+3)^{1/3)} ?

The integral of (3x)/((x^2+3)^{1/3)} is 9/4 (x^2+3)^{2/3}+C

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integral of (3x)/((x^2+3)^{1/3)}

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

\frac{9}{4} (x^{2} + 3)^{\frac{2}{3}} + C

Solution steps

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

Take the constant out:

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

= 3 \cdot \int \frac{x}{( x ^{2} + 3 ) ^{\frac{1}{3}}} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: \frac{3}{2} u^{\frac{2}{3}}

= 3 \cdot \frac{1}{2} \cdot \frac{3}{2} u^{\frac{2}{3}}

Substitute back

u = x^{2} + 3

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

Simplify

3 \cdot \frac{1}{2} \cdot \frac{3}{2} (x^{2} + 3)^{\frac{2}{3}} : \frac{9}{4} (x^{2} + 3)^{\frac{2}{3}}

= \frac{9}{4} (x^{2} + 3)^{\frac{2}{3}}

Add a constant to the solution

= \frac{9}{4} (x^{2} + 3)^{\frac{2}{3}} + C

\frac{d}{d x} (\frac{( 1 - x ^{2} )}{2 - x})

x \to 3 - lim (\frac{1}{∣ x - 3 ∣})

\int_{0}^{9} x d x

\frac{d}{d x} f (x) = (4 x^{(4)} + 6)^{(3)}

g (x) = 5 x^{2} - x + 7

What is the integral of (3x)/((x^2+3)^{1/3)} ?
The integral of (3x)/((x^2+3)^{1/3)} is 9/4 (x^2+3)^{2/3}+C