What is the integral of ((x+1))/(sqrt((x^2+2))) ?

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

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integral of ((x+1))/(sqrt((x^2+2)))

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

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

Solution steps

\int \frac{x + 1}{x ^{2} + 2} d x

Expand

\frac{x + 1}{x ^{2} + 2} : \frac{x}{x ^{2} + 2} + \frac{1}{x ^{2} + 2}

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

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

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

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

Add a constant to the solution

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

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