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

The integral of (2x+y+z)/((x+y)(x+z)) is ln(|y+x||z+x|)+C

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integral of (2x+y+z)/((x+y)(x+z))

\int \frac{2 x + y + z}{( x + y ) ( x + z )} d x

ln (∣ y + x ∣ ∣ z + x ∣) + C

Solution steps

\int \frac{2 x + y + z}{( x + y ) ( x + z )} d x

Apply u-substitution

: \int \frac{1}{u} d u

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = (y + x) (z + x)

= ln ∣ (y + x) (z + x) ∣

Simplify

ln ∣ (y + x) (z + x) ∣ : ln (∣ y + x ∣ ∣ z + x ∣)

= ln (∣ y + x ∣ ∣ z + x ∣)

Add a constant to the solution

= ln (∣ y + x ∣ ∣ z + x ∣) + C

8 x^{2} - 7 x, at 9

y^{3} - (6 x + 8) + 3 x y^{2} y^{^{'}} = 0

4 y^{^{''}} + 4 y^{^{'}} + 1 y = 0, y (2) = - 6, y^{^{'}} (2) = 4

\int_{3}^{6} \frac{x}{x - 3} d x

k \to \infty lim (\frac{1}{k})

What is the integral of (2x+y+z)/((x+y)(x+z)) ?
The integral of (2x+y+z)/((x+y)(x+z)) is ln(|y+x||z+x|)+C