Question

Evaluate integration 1÷(x+1)(x+4)dx ?

Answer 1

`I=1/3(ln(x+1)-ln(x+4))+C`

Explanation:

We want to integrate

`I=int1/((x+1)(x+4))dx`

By partial fraction decomposition

`I=intA/(x+1)+B/(x+4)dx`

Find the constants `A` and `B`

`1=A(x+4)+B(x+1)`

Let `x=-4` then `B=-1/3`
Let `x=-1` then `A=1/3`

Thus

`I=1/3int1/(x+1)-1/(x+4)dx`

`color(white)(I)=1/3(ln(x+1)-ln(x+4))+C`