Question

How do you use Integration by Substitution to find `intdx/(1-6x)^4dx`?

Answer 1

`int (1 - 6x)^(-4) dx =` ?

We will let `u = 1 - 6x`. Thus, `du = -6 dx`.

`= int u^(-4) dx`

This looks difficult since there isn't a `-6` in there for us to form a `du` with. However, there is a rule of integration which states:

`int c*f(x) dx = c * int f(x) dx`

We can exploit this rule to rewrite our integral equivalently as:

`= -1/6 int -6 u^(-4) dx`

The statements are completely equivalent; note that if we pull the `-6` out of the integral, then it cancels with the `-1/6` and leaves us with `1` multiplied by our original integral.

Anyway, we now have a `-6 dx` to form a `du` with.

`= -1/6 int u^(-4) du`

`= -1/6 u^(-3) * (-1/3)`

`= 1/18 u^(-3)`

`= 1/(18u^3)`

Substituting back for `u` gives us:

`= 1/(18(1 - 6x)^3)`