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

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

integral

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

\frac{x ^{2}}{2} + x + ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

Apply the Sum Rule:

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

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

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

\int 1 d x = x

\int \frac{1}{x} d x = ln ∣ x ∣

= \frac{x ^{2}}{2} + x + ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{2}}{2} + x + ln ∣ x ∣ + C

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

\frac{1}{( a ^{3} - 2 a )}

4 - ∣ x ∣

(9 x + 4)^{4}

cos (2 x + 3)

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