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

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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

(x^{4} - x + y) d x - x d y = 0

ln (x x^{2} + 1)

f (x) = 6 x - 6 x

\int_{1}^{2} x e^{x^{2}} d x

\frac{d y}{d x} = (4 y)

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