Question

In the evaluation of $\int \frac{dx}{\sqrt{-x^2-x+2}}$ using Euler's substitution, which of the following is correct?

• A. As the leading coefficient of the quadratic $a$ and the vertical translation have different signs, polynomial can be factorised using real roots, first Euler substitution is used.
• B. As the leading coefficient of the quadratic $a>0$, First Euler substitution is used.
• C. As the leading coefficient of the quadratic $a$ and the vertical translation have different signs, polynomial can be factorised using real roots, Second Euler substitution is used.
• D. As the leading coefficient of the quadratic $a>0$, Second Euler substitution is used.

Answer 1

The correct option is C As the leading coefficient of the quadratic $a$ and the vertical translation have different signs, polynomial can be factorised using real roots, Second Euler substitution is used.

If $c>0$ we take $\sqrt{a{x}^2+bx+c}=xt\pm \sqrt{c}.$

We solve for $x$ similarly as above and find, $x=\frac{\pm 2t\sqrt{c}-b}{a-t^2}.$

Again, either the positive or the negative sign can be chosen.

This happens in a Second Euler substitution.