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

The integral of (2x+3)/(sqrt(x+2)) is 2(2/3 (x+2)^{3/2}-sqrt(x+2))+C

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

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

2 (\frac{2}{3} (x + 2)^{\frac{3}{2}} - x + 2) + C

Solution steps

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

Apply u-substitution

: \int 2 (2 u^{2} - 1) d u

= \int 2 (2 u^{2} - 1) d u

Take the constant out:

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

= 2 \cdot \int 2 u^{2} - 1 d u

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 2 u^{2} d u - \int 1 d u)

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

\int 1 d u = u

= 2 (\frac{2 u ^{3}}{3} - u)

Substitute back

u = x + 2

= 2 (\frac{2 ( x + 2 ) ^{3}}{3} - x + 2)

Simplify

2 (\frac{2 ( x + 2 ) ^{3}}{3} - x + 2) : 2 (\frac{2}{3} (x + 2)^{\frac{3}{2}} - x + 2)

= 2 (\frac{2}{3} (x + 2)^{\frac{3}{2}} - x + 2)

Add a constant to the solution

= 2 (\frac{2}{3} (x + 2)^{\frac{3}{2}} - x + 2) + C

\frac{d}{d x} (sin (x^{2}) cos (π x + 2))

\frac{\partial}{\partial y} (4 x^{3} - 36 x)

\int 4 y e^{y^{2}} d y

(1999, 67%), (2005, 92%)

x^{5}, 1

What is the integral of (2x+3)/(sqrt(x+2)) ?
The integral of (2x+3)/(sqrt(x+2)) is 2(2/3 (x+2)^{3/2}-sqrt(x+2))+C