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

The integral of 1/(sqrt(x(x+1)^3)) is 2sqrt(x/(1+x))+C

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integral of 1/(sqrt(x(x+1)^3))

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

2 \frac{x}{1 + x} + C

Solution steps

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

x (x + 1)^{3} = x (x + 1)^{3}

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

Apply u-substitution

= \int \frac{2}{( u ^{2} + 1 ) ^{\frac{3}{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} + 1 ) ^{\frac{3}{2}}} d u

Apply Trigonometric Substitution

= 2 \cdot \int cos (v) d v

Use the common integral:

\int cos (v) d v = sin (v)

= 2 sin (v)

Substitute back

= 2 sin (arctan (x))

Simplify

2 sin (arctan (x)) : 2 \frac{x}{1 + x}

= 2 \frac{x}{1 + x}

Add a constant to the solution

= 2 \frac{x}{1 + x} + C

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