What is the integral of x/((5x^2-125)^{1/3)} ?

The integral of x/((5x^2-125)^{1/3)} is 3/20 (5x^2-125)^{2/3}+C

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integral of x/((5x^2-125)^{1/3)}

\int \frac{x}{( 5 x ^{2} - 125 ) ^{\frac{1}{3}}} d x

\frac{3}{20} (5 x^{2} - 125)^{\frac{2}{3}} + C

Solution steps

\int \frac{x}{( 5 x ^{2} - 125 ) ^{\frac{1}{3}}} d x

Apply u-substitution

: \int \frac{1}{10 u ^{\frac{1}{3}}} d u

= \int \frac{1}{10 u ^{\frac{1}{3}}} d u

Take the constant out:

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

= \frac{1}{10} \cdot \int \frac{1}{u ^{\frac{1}{3}}} d u

Apply the Power Rule

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

= \frac{1}{10} \cdot \frac{3}{2} u^{\frac{2}{3}}

Substitute back

u = 5 x^{2} - 125

= \frac{1}{10} \cdot \frac{3}{2} (5 x^{2} - 125)^{\frac{2}{3}}

Simplify

\frac{1}{10} \cdot \frac{3}{2} (5 x^{2} - 125)^{\frac{2}{3}} : \frac{3}{20} (5 x^{2} - 125)^{\frac{2}{3}}

= \frac{3}{20} (5 x^{2} - 125)^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{20} (5 x^{2} - 125)^{\frac{2}{3}} + C

y^{^{'}} = 0.0013 y (3000 - y)

2 y^{^{'}} + 9 y = 3, y (0) = 1

x \to 0 - lim (- \frac{1}{x ^{2}})

\frac{d}{d x} ((x + 6) 2)

\int_{0}^{3} \frac{1}{x ^{2}} d x

What is the integral of x/((5x^2-125)^{1/3)} ?
The integral of x/((5x^2-125)^{1/3)} is 3/20 (5x^2-125)^{2/3}+C