Question

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

• A. As the leading coefficient of the quadratic $a>0$, First Euler substitution is used.
• B. As the leading coefficient of the quadratic $a>0$, Second Euler substitution is used.
• C. As the leading coefficient of the quadratic $a>0$, third Euler substitution is used.
• D. None of these

Answer 1

Answer: (A)

As the leading coefficient of the quadratic $a>0$, First Euler substitution is used.

If leading co efficient of quadratic equation $a{x}^2+bx+c=0$ is greater than $0$, then first substitution ruler is used.
$\sqrt{a{x}^2+bx+c=0}=\pm x\sqrt{a}+t$
Here, $I=\int \frac{dx}{x+\sqrt{x^2+2x+2}}$
Here coefficient of $x^2=1>0$.
Therefore, first ruler substitution is used.