Question

does squaring a rational expression inside an integral affect dx?

Answer 1

I plan to do it on a problem that looks like this

Answer 2

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

Answer 3

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
\[I=\int\limits \frac{dt}{\sqrt{a^2-t^2}}=\sin^{-1}\frac{t}{a}\]

Answer 4

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