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

Concept: U Substitution — split the numerator to match the derivative of the denominator’s inside.

Step 1: Write the integral as a sum:

Step 2: For the first term, let $u = x^2 - 1$, so $du = 2x\,dx$ and $x\,dx = \frac{du}{2}$. Then

Step 3: The second term is a standard form: …

We are integrating $\frac{x+2}{\sqrt{x^2 - 1}}$. The denominator $\sqrt{x^2 - 1}$ suggests two classic patterns: the derivative of $\sqrt{x^2 - 1}$ is $\frac{x}{\sqrt{x^2 - 1}}$, and the derivative of $\log|x + \sqrt{x^2 - 1}|$ is $\frac{1}{\sqrt{x^2 - 1}}$. So the numerator $x+2$ is perfectly set up to be split into $x$ and $2$, each handled by one of these patterns.

Let’s work through it step by step.

1. Split the integrand Write the integral as the sum of two simpler integrals:

2. First integral: $\int \frac{x}{\sqrt{x^2 - 1}} \, dx$ Notice that the numerator $x$ is (up to a constant) the derivative of $x^2 - 1$, which sits inside the square root. This is a classic candidate for substitution. Let $u = x^2 - 1$. Then $du = 2x \, dx$, so $x \, dx = \frac{du}{2}$. The integral becomes:

Integrating: $\frac12 \cdot 2 u^{1/2} = \sqrt{u} = \sqrt{x^2 - 1}$.

So the first part gives $\sqrt{x^2 - 1}$.

3. Second integral: $2 \int \frac{1}{\sqrt{x^2 - 1}} \, dx$ This is a standard form. Recall that $\frac{d}{dx} \log|x + \sqrt{x^2 - 1}| = \frac{1}{\sqrt{x^2 - 1}}$. Therefore, $2 \int \frac{1}{\sqrt{x^2 - 1}} \, dx = 2 \log|x + \sqrt{x^2 - 1}|.$ …

Method: $\dfrac{\text{linear}}{\sqrt{\text{quadratic}}}$ — split the numerator

Write the linear numerator as (a multiple of the derivative of the radicand) + (a constant), so the integral splits into a $u$-substitution part and a standard-form part.

Steps

Step 1: Split the numerator. For $\displaystyle\int \frac{px+q}{\sqrt{Q(x)}}\,dx$ with $Q(x)=ax^2+bx+c$, write

and solve for $\lambda,\mu$ by matching coefficients.

Step 2: First piece — $u$-substitution.

$\lambda\int \frac{Q'(x)}{\sqrt{Q(x)}}\,dx = 2\lambda\sqrt{Q(x)} .$ …

Common Mistakes

Mistake 1: Integrating the numerator and denominator separately.

Why it's wrong: $\int \frac{x+2}{\sqrt{x^2-1}}\,dx \ne \frac{\int(x+2)}{\int\sqrt{x^2-1}}$. Correct approach: split the numerator into a derivative-part plus a constant, then integrate each piece.

Mistake 2: Losing the constant piece $\mu$. …