What is the integral of x^{-1}*1/(2+x^2) ?

The integral of x^{-1}*1/(2+x^2) is -1/4 (ln|x^2+2|-2ln(x))+C

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integral of x^{-1}*1/(2+x^2)

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

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

Solution steps

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

Simplify

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

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

Apply u-substitution

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

Complete the square

(u - 2) u : (u - 1)^{2} - 1

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

Apply u-substitution

= \frac{1}{2} \cdot \int \frac{1}{v ^{2} - 1} d v

Factor

v^{2} - 1 : - (- v^{2} + 1)

= \frac{1}{2} \cdot \int \frac{1}{- ( - v ^{2} + 1 )} d v

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= \frac{1}{2} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

What is the integral of x^{-1}*1/(2+x^2) ?
The integral of x^{-1}*1/(2+x^2) is -1/4 (ln|x^2+2|-2ln(x))+C