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

The integral of 3/(1+sqrt(x+1)) is 3(-2ln|sqrt(x+1)+1|+2sqrt(x+1)+2)+C

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

\int \frac{3}{1 + x + 1} d x

3 (- 2 ln x + 1 + 1 + 2 x + 1 + 2) + C

Solution steps

\int \frac{3}{1 + x + 1} d x

Take the constant out:

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

= 3 \cdot \int \frac{1}{1 + x + 1} d x

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

= 3 \cdot \int - \frac{x + 1}{1 + x + 1} + 1 d x

Apply the Sum Rule:

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

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

\int \frac{x + 1}{1 + x + 1} d x = x + 2 ln x + 1 + 1 - 2 x + 1 - 2

\int 1 d x = x

= 3 (- (x + 2 ln x + 1 + 1 - 2 x + 1 - 2) + x)

Simplify

= 3 (- 2 ln x + 1 + 1 + 2 x + 1 + 2)

Add a constant to the solution

= 3 (- 2 ln x + 1 + 1 + 2 x + 1 + 2) + C

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