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

The integral of (2x+3)/((2+sqrt(x))^2) is 2(x-8sqrt(x)+(22)/(sqrt(x)+2)+27ln|sqrt(x)+2|-20)+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)/((2+sqrt(x))^2)

\int \frac{2 x + 3}{( 2 + x ) ^{2}} d x

2 (x - 8 x + \frac{22}{x + 2} + 27 ln x + 2 - 20) + C

Solution steps

\int \frac{2 x + 3}{( 2 + x ) ^{2}} d x

Apply u-substitution

: \int \frac{2 ( u - 2 ) ( 2 u ^{2} - 8 u + 11 )}{u ^{2}} d u

= \int \frac{2 ( u - 2 ) ( 2 u ^{2} - 8 u + 11 )}{u ^{2}} 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 - 2 ) ( 2 u ^{2} - 8 u + 11 )}{u ^{2}} d u

Expand

\frac{( u - 2 ) ( 2 u ^{2} - 8 u + 11 )}{u ^{2}} : 2 u - 12 + \frac{27}{u} - \frac{22}{u ^{2}}

= 2 \cdot \int 2 u - 12 + \frac{27}{u} - \frac{22}{u ^{2}} 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 d u - \int 12 d u + \int \frac{27}{u} d u - \int \frac{22}{u ^{2}} d u)

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

\int 12 d u = 12 u

\int \frac{27}{u} d u = 27 ln ∣ u ∣

\int \frac{22}{u ^{2}} d u = - \frac{22}{u}

= 2 (u^{2} - 12 u + 27 ln ∣ u ∣ - (- \frac{22}{u}))

Substitute back

u = 2 + x

= 2 ((2 + x)^{2} - 12 (2 + x) + 27 ln 2 + x - (- \frac{22}{2 + x}))

Simplify

2 ((2 + x)^{2} - 12 (2 + x) + 27 ln 2 + x - (- \frac{22}{2 + x})) : 2 (x - 8 x + \frac{22}{x + 2} + 27 ln x + 2 - 20)

= 2 (x - 8 x + \frac{22}{x + 2} + 27 ln x + 2 - 20)

Add a constant to the solution

= 2 (x - 8 x + \frac{22}{x + 2} + 27 ln x + 2 - 20) + C

\int \frac{( x - 1 ) e ^{x^{2}}}{e ^{2 x}} d x

\int 9 x e^{0.005 x^{2}} d x

\int (4 x - 5)^{2} d x

\int \frac{14 x ^{3} + 2 x + 1}{x ^{3}} d x

\int (- \frac{3}{x}) d x

What is the integral of (2x+3)/((2+sqrt(x))^2) ?
The integral of (2x+3)/((2+sqrt(x))^2) is 2(x-8sqrt(x)+(22)/(sqrt(x)+2)+27ln|sqrt(x)+2|-20)+C