What is the integral of (4x^2-x+5)/(12x) ?

The integral of (4x^2-x+5)/(12x) is 1/12 (2x^2-x+5ln|x|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (4x^2-x+5)/(12x)

\int \frac{4 x ^{2} - x + 5}{12 x} d x

\frac{1}{12} (2 x^{2} - x + 5 ln ∣ x ∣) + C

Solution steps

\int \frac{4 x ^{2} - x + 5}{12 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{12} \cdot \int \frac{4 x ^{2} - x + 5}{x} d x

Expand

\frac{4 x ^{2} - x + 5}{x} : 4 x - 1 + \frac{5}{x}

= \frac{1}{12} \cdot \int 4 x - 1 + \frac{5}{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

= \frac{1}{12} (\int 4 x d x - \int 1 d x + \int \frac{5}{x} d x)

\int 4 x d x = 2 x^{2}

\int 1 d x = x

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= \frac{1}{12} (2 x^{2} - x + 5 ln ∣ x ∣)

Add a constant to the solution

= \frac{1}{12} (2 x^{2} - x + 5 ln ∣ x ∣) + C

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

\int_{0}^{4} (2 t + 4) d t

10 (4 e^{x^{2}} x^{3} + 6 e^{x^{2}} x)

x \to 0 lim (lo g_{10} (x))

\int \frac{1}{x ^{2} x ^{2} + 16} d x

What is the integral of (4x^2-x+5)/(12x) ?
The integral of (4x^2-x+5)/(12x) is 1/12 (2x^2-x+5ln|x|)+C