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

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

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

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

x^{2} + 4 + 2 ln (\frac{1}{2} x + 4 + x^{2}) + C

Solution steps

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

Expand

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

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

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

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

\int \frac{2}{x ^{2} + 4} d x = 2 ln (\frac{1}{2} x + 4 + x^{2})

= x^{2} + 4 + 2 ln (\frac{1}{2} x + 4 + x^{2})

Add a constant to the solution

= x^{2} + 4 + 2 ln (\frac{1}{2} x + 4 + x^{2}) + C

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