What is the integral of (4x+3)/(x^2+4) ?

The integral of (4x+3)/(x^2+4) is 2ln|x^2+4|+3/2 arctan(x/2)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (4x+3)/(x^2+4)

\int \frac{4 x + 3}{x ^{2} + 4} d x

2 ln x^{2} + 4 + \frac{3}{2} arctan (\frac{x}{2}) + C

Solution steps

\int \frac{4 x + 3}{x ^{2} + 4} d x

Expand

\frac{4 x + 3}{x ^{2} + 4} : \frac{4 x}{x ^{2} + 4} + \frac{3}{x ^{2} + 4}

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{4 x}{x ^{2} + 4} d x + \int \frac{3}{x ^{2} + 4} d x

\int \frac{4 x}{x ^{2} + 4} d x = 2 ln x^{2} + 4

\int \frac{3}{x ^{2} + 4} d x = \frac{3}{2} arctan (\frac{x}{2})

= 2 ln x^{2} + 4 + \frac{3}{2} arctan (\frac{x}{2})

Add a constant to the solution

= 2 ln x^{2} + 4 + \frac{3}{2} arctan (\frac{x}{2}) + C

What is the integral of (4x+3)/(x^2+4) ?
The integral of (4x+3)/(x^2+4) is 2ln|x^2+4|+3/2 arctan(x/2)+C