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

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

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

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

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

Solution steps

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

Apply u-substitution

= \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

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

Substitute back

u = 1 + x

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

Simplify

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

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

Add a constant to the solution

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

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