Question

How do i integrate this using substitution method (x*5+3)/(x*6+18x+3)?

Answer 1

`int(x^5+3)/(x^6+18x+3)dx=1/6ln|x^6+18x+3|+C`

Explanation:

So, we want

`int(x^5+3)/(x^6+18x+3)dx`

We really only have two choices for the substitution `u`. As a general rule (it's not foolproof), when making a substitution in a rational function, consider the denominator first:

`u=x^6+18x+3`

`du=(6x^5+18)dx`

This seems to resemble the numerator, but larger. However, dividing both sides by `6` yields

`1/6du=(6x^5+18)/6dx`

`1/6du=(x^5+3)dx`

So, this substitution is indeed valid.

Apply the substitution:

`1/6int(du)/u=1/6ln|u|+C`

Rewrite in terms of `x:`

`int(x^5+3)/(x^6+18x+3)dx=1/6ln|x^6+18x+3|+C`