What is the integral of ((4x+6)/(x^2+4)) ?

The integral of ((4x+6)/(x^2+4)) is 2ln|x^2+4|+3arctan(x/2)+C

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integral of ((4x+6)/(x^2+4))

\int (\frac{4 x + 6}{x ^{2} + 4}) d x

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

Solution steps

\int \frac{4 x + 6}{x ^{2} + 4} d x

Expand

\frac{4 x + 6}{x ^{2} + 4} : \frac{4 x}{x ^{2} + 4} + \frac{6}{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{6}{x ^{2} + 4} d x

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

\int \frac{6}{x ^{2} + 4} d x = 3 arctan (\frac{x}{2})

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

Add a constant to the solution

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

x = 0, x = 5 - y^{2}

\frac{8}{( s + 2 ) ^{2}}

9 y^{^{''}} + 60 y^{^{'}} + 100 y = 0

8 (x + 2)^{\frac{3}{2}}

x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}

What is the integral of ((4x+6)/(x^2+4)) ?
The integral of ((4x+6)/(x^2+4)) is 2ln|x^2+4|+3arctan(x/2)+C