What is the integral of (x+3)/(x^2+2x+1) ?

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

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

ln ∣ x + 1 ∣ - \frac{2}{x + 1} + C

Solution steps

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

Expand

\frac{x + 3}{x ^{2} + 2 x + 1} : \frac{x}{x ^{2} + 2 x + 1} + \frac{3}{x ^{2} + 2 x + 1}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{x}{x ^{2} + 2 x + 1} d x = ln ∣ x + 1 ∣ + \frac{1}{x + 1}

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

= ln ∣ x + 1 ∣ + \frac{1}{x + 1} - \frac{3}{x + 1}

Simplify

ln ∣ x + 1 ∣ + \frac{1}{x + 1} - \frac{3}{x + 1} : ln ∣ x + 1 ∣ - \frac{2}{x + 1}

= ln ∣ x + 1 ∣ - \frac{2}{x + 1}

Add a constant to the solution

= ln ∣ x + 1 ∣ - \frac{2}{x + 1} + C

\int x + \frac{1}{8 x} d x

\int 18 (3 x + 2)^{2} d x

\int \frac{5 x - 3}{x ^{2} + 8 x + 7} d x

\int 12 x^{- 1} d x

\int (sec^{2} (t) i + \frac{1}{1 + t ^{2}} j) d t

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