1
$\begingroup$

Im trying to integrate $ \int\frac{2x-\sqrt{4x^{2}-x+1}}{x-1}dx $.

I've tried integration by parts, and any resonable substitution that came to my mind. Neither seem to work. I tried to use integral online calculator, the result was too complicated, and I'm pretty sure there is a resonable way to solve it, as we had this question in an exam.

Thanks in advance.

$\endgroup$
3
  • $\begingroup$ Hint: $(2x-\sqrt{4x^{2}-x+1})(2x+\sqrt{4x^{2}-x+1})=x-1$ $\endgroup$ – Eminem Aug 15 '20 at 12:21
  • $\begingroup$ It's not an impossible integral but the solution is not particularly simple for an exam question. $\endgroup$ – Peter Foreman Aug 15 '20 at 12:22
  • $\begingroup$ First thing I tried was the hint suggested in the first comment, it did not lead me to the solution. Can someone point out a way that leads to the solution ? (just name the steps and I'll do it myself) $\endgroup$ – FreeZe Aug 15 '20 at 12:26
2
$\begingroup$

Take,$$t=\sqrt{4x^2-x+1}-2x$$

Then,$$\sqrt{4x^2-x+1}=2x+t \;\;\Longrightarrow\;\;4x^2-x+1=(2x+t)^2 \;\;\Longrightarrow\;\; x=\frac{1-t^2}{4t+1}\,.$$

And $$dx = -\frac{4t^2+2t+4}{(4t+1)^2}dt\,.$$

So,

$$I=\int\frac{2x-\sqrt{4x^2-x+1}}{x-1}dx=\int\frac{t}{(1-t^2)/(4t+1)-1}\cdot\frac{4t^2+2t+4}{(4t+1)^2}dt\,,$$

$$\begin{align} & =-\int\frac{4t^2+2t+4}{(4t+1)(t+4)}dt\\ &=-\int\frac{4t^2+2t+4}{4t^2+17t+4}dt\\ &=-\int\left(1-\frac{15t}{4t^2+17t+4}\right)dt\\ &=-t+\int\frac{15t}{(4t+1)(t+4)}dt\,. \end{align}$$

Splitting into partial fractions,

$$\frac{15t}{(4t+1)(t+4)} = \frac{4}{t+4}-\frac{1}{4t+1}\,,$$

We get,

$$I=-t+\int\frac{4}{t+4}dt-\int\frac{1}{4t+1}dt=-t+4\ln(t+4)-\frac{1}{4}\ln(4t+1)+C\,,$$

so, therefore,

$$\Rightarrow \bbox[5px,border:2px solid red]{I=-t+4\ln(t+4)-\frac{1}{4}\ln(4t+1)+C\,,}$$

On rewriting we get,

$$\Rightarrow \bbox[5px,border:2px solid red]{\begin{align}I\;=\;&2x-\sqrt{4x^2-x+1}+4\ln(\sqrt{4x^2-x+1}-2x+4)\\ &-\frac{1}{4}\ln(4\sqrt{4x^2-x+1}-8x+1)+C\,. \end{align}}$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.