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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

\int \frac{64}{x ^{2} + 16} d x

\int \frac{[ ln ( x ) ] ^{2}}{x} d x

\int 2 x^{3} e^{- x} d x

\int \frac{x ^{4}}{7 x ^{5} + 1} d x

\int (x + \frac{9}{2}) e^{x^{2} + 9 x + 1} d x

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