What is the integral of 1/(sqrt(x)(1+\sqrt{x))^5} ?

The integral of 1/(sqrt(x)(1+\sqrt{x))^5} is -1/(2(sqrt(x)+1)^4)+C

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integral of 1/(sqrt(x)(1+\sqrt{x))^5}

\int \frac{1}{x ( 1 + x ) ^{5}} d x

- \frac{1}{2 ( x + 1 ) ^{4}} + C

Solution steps

\int \frac{1}{x ( 1 + x ) ^{5}} d x

Apply u-substitution

: \int \frac{2}{( 1 + u ) ^{5}} d u

= \int \frac{2}{( 1 + u ) ^{5}} 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}{( 1 + u ) ^{5}} d u

Apply u-substitution

: \int \frac{1}{v ^{5}} d v

= 2 \cdot \int \frac{1}{v ^{5}} d v

Apply the Power Rule

: - \frac{1}{4 v ^{4}}

= 2 (- \frac{1}{4 v ^{4}})

Substitute back

= 2 (- \frac{1}{4 ( 1 + x ) ^{4}})

Simplify

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

= - \frac{1}{2 ( x + 1 ) ^{4}}

Add a constant to the solution

= - \frac{1}{2 ( x + 1 ) ^{4}} + C

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What is the integral of 1/(sqrt(x)(1+\sqrt{x))^5} ?
The integral of 1/(sqrt(x)(1+\sqrt{x))^5} is -1/(2(sqrt(x)+1)^4)+C