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

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

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

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

\frac{2}{- x + 2} + C

Solution steps

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

Factor

x - 2 : - (- x + 2)

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

(- (- x + 2))^{2} = (- x + 2)^{2}

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

Apply u-substitution

: \int - \frac{2}{u ^{2}} d u

= \int - \frac{2}{u ^{2}} d u

Take the constant out:

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

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

Apply the Power Rule

: - \frac{1}{u}

= - 2 (- \frac{1}{u})

Substitute back

u = - x + 2

= - 2 (- \frac{1}{- x + 2})

Simplify

- 2 (- \frac{1}{- x + 2}) : \frac{2}{- x + 2}

= \frac{2}{- x + 2}

Add a constant to the solution

= \frac{2}{- x + 2} + C

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