What is the integral of 1/(sqrt(x+2)+3) ?

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

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integral of 1/(sqrt(x+2)+3)

\int \frac{1}{x + 2 + 3} d x

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

Solution steps

\int \frac{1}{x + 2 + 3} d x

Apply u-substitution

= \int \frac{2 u}{u + 3} d u

Take the constant out:

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

= 2 \cdot \int \frac{u}{u + 3} d u

Apply u-substitution

= 2 \cdot \int \frac{v - 3}{v} d v

Expand

\frac{v - 3}{v} : 1 - \frac{3}{v}

= 2 \cdot \int 1 - \frac{3}{v} d v

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 1 d v - \int \frac{3}{v} d v)

\int 1 d v = v

\int \frac{3}{v} d v = 3 ln ∣ v ∣

= 2 (v - 3 ln ∣ v ∣)

Substitute back

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

Add a constant to the solution

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

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