Improper Integral of $\int\frac{dx}{(2x-1)^3}$

Question

Improper Integral of

$$\int_{-\infty}^0\frac{dx}{(2x-1)^3}$$

from Anton Calculus 8th Edition, page 576, question 9. Answer is $-\frac{1}{4}$ but I'm finding $-1$

The integral, substituting $u= (2x-1)$:

$$\frac{1}{2}\cdot\frac{-2}{(2x-1)^2}+C$$

Definite solution to use with limit:

$$ \frac{1}{2}\cdot\frac{-2}{(2x-1)^2)}\Big|^0_a = \frac{1}{2}\cdot\left(-2+\frac{2}{(2a-1)^2}\right)$$

Then solving limit:

$$\lim_{x \to {-\infty}} \frac{1}{2}\cdot\left(-2+\frac{2}{(2a-1)^2}\right) = \frac{1}{2}\cdot(-2+0) = -1$$

Any leads?

Thank you.

Answer 1

You made correct substitution $$u= 2x-1\iff du=2dx$$ $$\int\frac{dx}{(2x-1)^3}dx=\frac12\int\frac{du}{u^3}dx=-\frac{1}{4u^2}+C$$

$$\int\frac{dx}{(2x-1)^3}=-\frac{1}{4(2x-1)^2}+C$$

I hope you can take it from here!

Answer 2

Rather than multiplying by $-2,$ you should be dividing, since $$\int u^{n}\,du=\frac1{n+1}u^{n+1}$$ for all integers $n\ne-1.$