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

The integral of (x^2+1)/(x^2+2) is -sqrt(2)arctan(x/(sqrt(2)))+x+1/(sqrt(2))arctan(x/(sqrt(2)))+C

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

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

- 2 arctan (\frac{x}{2}) + x + \frac{1}{2} arctan (\frac{x}{2}) + C

Solution steps

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

Expand

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

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

\int \frac{x ^{2}}{x ^{2} + 2} d x = - 2 arctan (\frac{x}{2}) + x

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

= - 2 arctan (\frac{x}{2}) + x + \frac{1}{2} arctan (\frac{x}{2})

Add a constant to the solution

= - 2 arctan (\frac{x}{2}) + x + \frac{1}{2} arctan (\frac{x}{2}) + C

x \to 3 - lim (\frac{7 - x}{x - 3})

\int \frac{x}{x ^{4} - 1} d x

\frac{\partial}{\partial x} (x^{2} y - y^{2} + y^{3} x^{2})

\int 1 - 4 x d x

cot^{2} (sin (θ))

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