What is the integral from 0 to 3 of 1/(1+sqrt(1+x)) ?

The integral from 0 to 3 of 1/(1+sqrt(1+x)) is 2ln(2)+2-2ln(3)

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integral from 0 to 3 of 1/(1+sqrt(1+x))

\int_{0}^{3} \frac{1}{1 + 1 + x} d x

2 ln (2) + 2 - 2 ln (3)

Decimal

1.18906 \dots

Solution steps

\int_{0}^{3} \frac{1}{1 + 1 + x} d x

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

= \int_{0}^{3} - \frac{x + 1}{1 + 1 + x} + 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

= - \int_{0}^{3} \frac{x + 1}{1 + 1 + x} d x + \int_{0}^{3} 1 d x

\int_{0}^{3} \frac{x + 1}{1 + 1 + x} d x = 2 ln (3) - 2 ln (2) + 1

\int_{0}^{3} 1 d x = 3

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

Simplify

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

What is the integral from 0 to 3 of 1/(1+sqrt(1+x)) ?
The integral from 0 to 3 of 1/(1+sqrt(1+x)) is 2ln(2)+2-2ln(3)