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

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

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

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

2 arctan (1 + x) + C

Solution steps

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

(x + 1)^{3} = (x + 1)^{\frac{3}{2}},

assuming

(x + 1) \geq 0

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

Factor

\frac{1}{x + 1 + ( x + 1 ) ^{\frac{3}{2}}}

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

Apply u-substitution

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

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

Use the common integral:

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

= 2 arctan (u)

Substitute back

u = 1 + x

= 2 arctan (1 + x)

Add a constant to the solution

= 2 arctan (1 + x) + C

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