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

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

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

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

- \frac{1}{( x + 1 ) ^{2}} + C

Solution steps

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

Apply u-substitution

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

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

Apply the Power Rule

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

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

Substitute back

u = x + 1

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

Simplify

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

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (4 x + 4 y)

\frac{\partial ^{2}}{\partial u ( w ) \partial w} (e^{u (w) - w})

\int \frac{1}{( x - 1 ) ( x ^{2} + 9 )} d x

\int \frac{1}{9 - 4 x} d x

y = \frac{- 12}{4 x}

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