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

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

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integral of x/((x+1)-sqrt(x+1))

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

x + 1 + 2 x + 1 + C

Solution steps

\int \frac{x}{x + 1 - x + 1} d x

Apply u-substitution

: \int 2 (u + 1) d u

= \int 2 (u + 1) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int u + 1 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int u d u + \int 1 d u)

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

\int 1 d u = u

= 2 (\frac{u ^{2}}{2} + u)

Substitute back

u = x + 1

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

Simplify

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

= x + 1 + 2 x + 1

Add a constant to the solution

= x + 1 + 2 x + 1 + C

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