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

1 Answer
Aug 6, 2014

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)