Question

In the evaluation of $\int \frac{dx}{(x+1)\sqrt{-x^2+x+1}}$ using Euler's substitution, which of the following is correct?

• A. Since constant of quadratic is greater than $0$, first euler substitution is used.
• B. Since constant of quadratic is greater than $0$, second euler substitution is used.
• C. Since constant of quadratic is greater than $0$, third euler substitution is used.
• D. None of these

Answer 1

The correct option is C Since constant of quadratic is greater than $0$, third euler substitution is used.

If the polynomial $a{x}^2+bx+c$ has real roots $\alpha$ and $\beta$ we may chose $\sqrt{a{x}^2+bx+c}=\sqrt{a(x-\alpha )(x-\beta )}=(x-\alpha )t$ this yield $x=\frac{\alpha \beta -\alpha t^2}{\alpha -t^2}$ and as in the preceding cases, we can express the entire integrated rationally via $t$.

Third euler substitution works this way.