Question

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

∫-6/x(4x+6)^2 dx.

Answer 1

`(1)/(6) (-(3)/((2x+3)) -\ln x+\ln(2x+3))+C`

Explanation:

We are given:
`\int(-6)/(x(4x+6)^2) dx`

`x = (3)/(2)\tan t\\dx =  (3)/(2)(\tan ^2 t +1) dt`

`\int(-6)/(x(4x+6)^2) dx= (1)/(6) \int(-(6)/((3+2x)^2) -(1)/(x) +(2)/(3+2x))dx=\\\\= (1)/(6) (-(3)/((2x+3)) -\ln x+\ln(2x+3))+C`