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

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

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

integral

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

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

Solution steps

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

Apply Integral Substitution

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{u ^{2} + 1} d u

Use the common integral:

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

= \frac{1}{2} arctan (u)

Substitute back

u = \frac{x}{2}

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

Add a constant to the solution

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

\frac{x ^{2}}{( x ^{3} + 1 )}

\frac{x}{( x + 2 )}

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

\frac{1}{( x ^{3} + x )}

(e^{- t})^{2}

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