## mpj4 one year ago does squaring a rational expression inside an integral affect dx?

1. mpj4

|dw:1433407624358:dw|I plan to do it on a problem that looks like this

2. mpj4

3. anonymous

Try something like $I=- \int\limits \frac{-x^2dx}{\sqrt{-x^2-4x+5}}$$I=-(\int\limits \frac{(-x^2-4x+5+4x-5)dx}{\sqrt{-x^2-4x+5}})$$I=-(\int\limits \frac{(-x^2-4x+5)dx}{\sqrt{-x^2-4x+5}}+4\int\limits \frac{xdx}{\sqrt{-x^2-4x+5}}-5\int\limits \frac{dx}{\sqrt{-x^2-4x+5}})$

4. anonymous

First integral factorize in the form of $a^2-(x+z)^2$ substitute x+z so it becomes $a^2-t^2$ Use the formula $\int\limits \sqrt{a^2-t^2} dt=\frac{t \sqrt{a^2-t^2}}{2}+\frac{a^2}{2}\sin^{-1}(\frac{t}{a})$ Second integral Let $x=A \frac{d(-x^2-4x+5)}{dx}+B$ Equate coefficients and solve Third Integral Factorize and apply $I=\int\limits \frac{dt}{\sqrt{a^2-t^2}}=\sin^{-1}\frac{t}{a}$

5. mpj4

So sorry, I was solving other problems and skipped this one. Thank you so much