Question

Solve: $\int \frac{dx}{\sqrt{x^2+2x+5}}$

• A. $\ln |\sqrt{x^2+2x+5}-x+1|+C$
• B. $\ln |\sqrt{x^2+2x+5}+x+1|+C$
• C. $\ln |\sqrt{x^2+2x+5}-x|+C$
• D. None of these

Answer 1

Answer: (B)

$\ln |\sqrt{x^2+2x+5}+x+1|+C$

We know that $\int \frac{dx}{\sqrt{x^2+a^2}}=\ln (x+\sqrt{x^2+a})+C$ ...... $\left(i\right)$

Let $I=\int \frac{dx}{\sqrt{x^2+2x+5}}$

Taking only the denominator part and split $5$ as $1+4$

$\Rightarrow I=\int \frac{dx}{\sqrt{x^2+2x+1+4}}$

$\Rightarrow I=\int \frac{dx}{\sqrt{(x+1{)}^2+2^2}}$

Using $\left(i\right)$, we get

$I=\ln |x+1+\sqrt{x^2+2x+5}|+C$