`int (x+2)/(x^2+5x) dx` Use partial fractions to find the indefinite integral

`int (x+2)/(x^2+5x) dx`

To solve using partial fraction method, the denominator of the integrand should be factored.

`(x+2)/(x^2+5x) = (x + 2)/(x(x+5))`

Then, express it as sum of fractions.

`(x+2)/(x(x+5)) = A/x + B/(x +5)`

To determine the values of A and B, multiply both sides by the LCD of the fractions present.

`x(x+5)*(x+2)/(x(x+5)) = (A/x + B/(x +5))*x(x+5)`

`x+2=A(x+5)+Bx`

Then, assign values to x in which either x or x+5 will become zero.

So,...

`int (x+2)/(x^2+5x) dx`

To solve using partial fraction method, the denominator of the integrand should be factored.

`(x+2)/(x^2+5x) = (x + 2)/(x(x+5))`

Then, express it as sum of fractions.

`(x+2)/(x(x+5)) = A/x + B/(x +5)`

To determine the values of A and B, multiply both sides by the LCD of the fractions present.

`x(x+5)*(x+2)/(x(x+5)) = (A/x + B/(x +5))*x(x+5)`

`x+2=A(x+5)+Bx`

Then, assign values to x in which either x or x+5 will become zero.

So, plug-in x=0 to get the value of A.

`0+2=A(0+5)+B(0)`

`2=5A`

`2/5=A`

Also, plug-in x=-5 to get the value of B.

`-5+2=A(-5+5)+B(-5)`

`-3=A(0)+B(-5)`

`-3=-5B`

`3/5=B`

So the partial fraction decomposition of the integrand is:

`int (x+2)(x^2+5x)dx`

`=int (x+2)/(x(x+5))dx`

`= int (2/(5x) +3/(5(x+5)))dx`

Then, express it as sum of two integrals.

`= int 2/(5x)dx + int 3/(5(x+5))dx`

`= 2/5 int 1/xdx + 3/5 int 1/(x+5)dx`

To take the integral of this, apply the formula `int 1/u du = ln|u|+C` .

`=2/5ln|x| + 3/5ln|x+5| + C`

Therefore, `int (x+2)/(x^2+5x)==2/5ln|x| + 3/5ln|x+5| + C` .