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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

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