Question

Evaluate the indefinite integral of 1/((2x - 1)sqrt(1-2x))

Answer 1

\[\int\limits \frac{1}{(2x - 1)\sqrt{1 - 2x}}\]

I know that I should use a change of variables replacing \[\sqrt{1 - 2x}\] with u and apply it's derivative to reach an easier integral, but I can't seem to rework the derivative of u to cancel any terms. I know the answer but need help with application of the derivative of u.

Answer 2

Two important suggestions:

1) Change the denominator to (1-2x) and include a minus 1 in the numerator (as they would be equivalent

2) The denominator now would have (1-2x) sqrt(1-2x) =
(1-2x)^(3/2)

So your integral now reads - (1-2x)^(-3/2) dx

which is a straightforward integral; let u = 1-2x.

So what we have done is to rewrite the entire expression.

Answer 3

No need to change any variables, etc.

Answer 4

Perfect! Thank you for your insight.

Answer 5

Very welcome.