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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

: \int \frac{v - 3}{v} d v

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

Add a constant to the solution

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

y^{^{''}} - 2 y^{^{'}} - 3 y = 64 x e^{- x}

\int_{0}^{a} (\frac{m}{2 π ( a - x )})^{5} d x

y = 3 x^{4} - 3 x^{2}, y = 3 x^{2}

x \to 2 lim (\frac{2 x + 4}{x - 7})

\int_{0}^{\infty} e^{- 0.2 x} d x

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