Q.Integrate the following function: $\frac{1}{\sqrt{x^2 + 2x + 2}}$

Concept: Hyperbolic Integration – completing the square to match the standard form $\int \frac{dx}{\sqrt{x^2 + a^2}} = \sinh^{-1}\frac{x}{a} + C$.

Step 1: Complete the square inside the square root:

Step 2: Substitute $u = x+1$, so $du = dx$. The integral becomes:

Step 3: This is the standard hyperbolic sine inverse form: …

Why Hyperbolic Substitution Works Here

When you see a quadratic inside a square root — especially one like $x^2 + 2x + 2$ that doesn't factor nicely — your first instinct might be trigonometric substitution. But there's a cleaner path.

The expression $\sqrt{x^2 + a^2}$ is the signature of a hyperbolic substitution. Why? Because the identity $\cosh^2 u - \sinh^2 u = 1$ lets us handle sums of squares without the messy sign changes that trig substitutions sometimes bring. More directly, the derivative of $\sinh^{-1} x$ is $\frac{1}{\sqrt{x^2 + 1}}$, so if we can force the integrand into that shape, the answer writes itself.

The real trick is to complete the square first. That turns a messy quadratic into a clean $(x+1)^2 + 1$, which is exactly $\text{(something)}^2 + 1^2$. Then a single substitution gives us the inverse hyperbolic sine.

Let's walk through it.

Step-by-Step Solution

1. Complete the square inside the radical.

The denominator is $\sqrt{x^2 + 2x + 2}$. Focus on the quadratic:

$x^2 + 2x + 2 = (x^2 + 2x + 1) + 1 = (x+1)^2 + 1$.

So the integral becomes:

2. Make a simple linear substitution.

Let $u = x + 1$. Then $du = dx$, and the integral is:

This is now a standard form. No need for a second substitution — we can integrate directly.

Here $a = 1$, so: …

Method: Complete the square, then use $\displaystyle\int\frac{dx}{\sqrt{u^2+a^2}}$

A quadratic under a square root that does not factor is handled by completing the square, converting it to $\sqrt{(\text{linear})^2\pm a^2}$, then applying a standard form.

Steps

Step 1: Complete the square inside the radical.

Step 2: Substitute the linear block (or read off directly).

With $u=x+1$, $du=dx$, the integral is $\displaystyle\int\frac{du}{\sqrt{u^2+1}}$.

Step 3: Apply the standard result. …

Common Mistakes

Mistake 1: Trying to integrate $\dfrac{1}{\sqrt{x^2+2x+2}}$ without completing the square.

Why it's wrong: the quadratic under the root does not match any standard form until it is written as a perfect square plus a constant. Correct approach: complete the square, $x^2+2x+2=(x+1)^2+1$.

Mistake 2: Mis-completing the square (e.g. $(x+2)^2$ or wrong constant). …