What is the integral from 2 to 7 of 1/(sqrt(x+2)+1) ?

The integral from 2 to 7 of 1/(sqrt(x+2)+1) is 2ln(3)+2-4ln(2)

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integral from 2 to 7 of 1/(sqrt(x+2)+1)

\int_{2}^{7} \frac{1}{x + 2 + 1} d x

2 ln (3) + 2 - 4 ln (2)

Decimal

1.42463 \dots

Solution steps

\int_{2}^{7} \frac{1}{x + 2 + 1} d x

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

= \int_{2}^{7} - \frac{x + 2}{x + 2 + 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

= - \int_{2}^{7} \frac{x + 2}{x + 2 + 1} d x + \int_{2}^{7} 1 d x

\int_{2}^{7} \frac{x + 2}{x + 2 + 1} d x = 4 ln (2) - 2 ln (3) + 3

\int_{2}^{7} 1 d x = 5

= - (4 ln (2) - 2 ln (3) + 3) + 5

Simplify

= 2 ln (3) + 2 - 4 ln (2)

What is the integral from 2 to 7 of 1/(sqrt(x+2)+1) ?
The integral from 2 to 7 of 1/(sqrt(x+2)+1) is 2ln(3)+2-4ln(2)