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

x^{4} cos (x^{3})

\frac{\partial}{\partial x} (4 cos (x^{2} + y^{2}))

e^{t} cos (6 t)

5 - x^{2}

f (x) = ln (x^{2} + 3)

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