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

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

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

\frac{( x + 1 ) ^{2}}{2} - 2 (x + 1) + 3 ln ∣ x + 1 ∣ + C

Solution steps

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

Apply u-substitution

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

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

Expand

\frac{u ^{2} - 2 u + 3}{u} : u - 2 + \frac{3}{u}

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

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

\int 2 d u = 2 u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

= \frac{u ^{2}}{2} - 2 u + 3 ln ∣ u ∣

Substitute back

u = x + 1

= \frac{( x + 1 ) ^{2}}{2} - 2 (x + 1) + 3 ln ∣ x + 1 ∣

Add a constant to the solution

= \frac{( x + 1 ) ^{2}}{2} - 2 (x + 1) + 3 ln ∣ x + 1 ∣ + C

\int θ e^{θ} d θ

\int x^{2} (5 x + 2 x^{3})^{8} d x

\int \frac{4}{x + 7} d x

\int (x^{2} + 1) \cdot sin (x) d x

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

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