What is the integral of (x^2)/(5+x) ?

The integral of (x^2)/(5+x) is ((5+x)^2)/2-10(5+x)+25ln|5+x|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^2)/(5+x)

\int \frac{x ^{2}}{5 + x} d x

\frac{( 5 + x ) ^{2}}{2} - 10 (5 + x) + 25 ln ∣ 5 + x ∣ + C

Solution steps

\int \frac{x ^{2}}{5 + x} d x

Apply u-substitution

: \int \frac{( u - 5 ) ^{2}}{u} d u

= \int \frac{( u - 5 ) ^{2}}{u} d u

Expand

\frac{( u - 5 ) ^{2}}{u} : u - 10 + \frac{25}{u}

= \int u - 10 + \frac{25}{u} 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

= \int u d u - \int 10 d u + \int \frac{25}{u} d u

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

\int 10 d u = 10 u

\int \frac{25}{u} d u = 25 ln ∣ u ∣

= \frac{u ^{2}}{2} - 10 u + 25 ln ∣ u ∣

Substitute back

u = 5 + x

= \frac{( 5 + x ) ^{2}}{2} - 10 (5 + x) + 25 ln ∣ 5 + x ∣

Add a constant to the solution

= \frac{( 5 + x ) ^{2}}{2} - 10 (5 + x) + 25 ln ∣ 5 + x ∣ + C

\int 13 lo g_{15} (x) d x

\int (x^{2} + x) \cdot e^{2 x} d x

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

x \to 0 + lim ((tan (6 x))^{x})

\frac{d y}{d x} = x (x - 1)

What is the integral of (x^2)/(5+x) ?
The integral of (x^2)/(5+x) is ((5+x)^2)/2-10(5+x)+25ln|5+x|+C