Question

Use the table of integrals, or a computer or calculator with symbolic integration capabilities, to find the indefinite integral.

∫2/3x(3x-5) dx.

Answer 1

`(2)/(3)((1)/(5)ln|3x-5|-(1)/(5)ln|x|)+C`

Explanation:

We have been given a indefinite integral
`\int (2)/(3x\left(3x-5\right))dx`. We are asked to find the indefinite integral.

We will use partial fraction formula to solve our given problem.

`(2)/(3x\left(3x-5\right))=(3)/(5(3x-5))-(1)/(5x)`

`\int (2)/(3x\left(3x-5\right))dx=(2)/(3)\int (1)/(x\left(3x-5\right))dx`

`(2)/(3)\int (1)/(x\left(3x-5\right))dx=(2)/(3)\int (3)/(5(3x-5))-(1)/(5x)dx`

Using difference rule of integrals, we will get:

`(2)/(3)(\int (3)/(5(3x-5))dx-\int (1)/(5x)dx)`

Now, we need to use u-substitution as:

Let
`u=3x-5`.

`(du)/(dx)=3`

`dx=(1)/(3)du`

`\int (3)/(5(3x-5))dx= (3)/(5)\int (1)/((u))*(1)/(3)du=(3)/(5)*(1)/(3)\int (1)/((u))du=(1)/(5)ln|u|=(1)/(5)ln|3x-5|`

`\int (1)/(5x)dx=(1)/(5)\int (1)/(x)dx=(1)/(5)ln|x|`

Substitute back these values:

`(2)/(3)(\int (3)/(5(3x-5))dx-\int (1)/(5x)dx)=(2)/(3)((1)/(5)ln|3x-5|-(1)/(5)ln|x|)`

Let us add a constant C.

`(2)/(3)((1)/(5)ln|3x-5|-(1)/(5)ln|x|)+C`

Therefore, our required integral would be
`(2)/(3)((1)/(5)ln|3x-5|-(1)/(5)ln|x|)+C`.