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

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

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integral of 1/(sqrt(x)(\sqrt{x)-2)^3}

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

- \frac{1}{4} x + C

Solution steps

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

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

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

Take the constant out:

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

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

\frac{1}{( - 2 ) ^{3}} = - \frac{1}{8}

= (- \frac{1}{8}) \cdot \int x^{- \frac{1}{2}} d x

Apply the Power Rule

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

= (- \frac{1}{8}) \cdot 2 x^{\frac{1}{2}}

Simplify

(- \frac{1}{8}) \cdot 2 x^{\frac{1}{2}} : - \frac{1}{4} x

= - \frac{1}{4} x

Add a constant to the solution

= - \frac{1}{4} x + C

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