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

The integral of x/(pix+2) is 1/(pi^2)(pix+2-2ln|pix+2|)+C

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integral of x/(pix+2)

\int \frac{x}{π x + 2} d x

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

Solution steps

\int \frac{x}{π x + 2} d x

Apply u-substitution

: \int \frac{u - 2}{π ^{2} u} d u

= \int \frac{u - 2}{π ^{2} u} d u

Take the constant out:

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

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

Expand

\frac{u - 2}{u} : 1 - \frac{2}{u}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= \frac{1}{π ^{2}} (u - 2 ln ∣ u ∣)

Substitute back

u = π x + 2

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

Add a constant to the solution

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

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