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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

f (x) = \frac{2}{3 x + 1}

5 sin (t)

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

a y^{^{'}} = b x - b y

\frac{d y}{d x} + (5 x) y = 3 x + 5

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