Q.(a) Find: $\displaystyle\int \sqrt{\dfrac{x+2}{x-2}}\,dx$.

Rationalizing the Denominator

Rationalizing the denominator means rewriting a fraction so that no radical (square root, cube root, …) is left on the bottom. It is algebraic housekeeping — the fraction's value never changes, because you only ever multiply by a cleverly disguised form of $1$.

Why bother? A quotient like $\dfrac{1}{\sqrt2}$ is awkward to estimate ($1\div 1.414$), but the equal form $\dfrac{\sqrt2}{2}$ is easy ($1.414\div2\approx0.707$). Cleaner denominators are also easier to add, compare and simplify, and most answer keys expect this final form.

Case 1 — a single square root

Multiply top and bottom by that root:

because $\sqrt5\times\sqrt5=5$ is rational. In general $\dfrac{a}{\sqrt b}=\dfrac{a\sqrt b}{b}.$

Case 2 — a sum or difference with a root

Here multiplying by the root alone fails; use the conjugate, which turns the denominator into a difference of squares:

Partial Fraction Decomposition

Adding $\frac{2}{x-1}+\frac{3}{x+2}$ over a common denominator gives $\frac{5x+1}{x^2+x-2}$. Partial fraction decomposition reverses this — it breaks one complicated rational function back into a sum of simple pieces. Those pieces are far easier to integrate: $\int \frac{2}{x-1}\,dx = 2\log|x-1|$ is immediate, while the combined fraction is not.

When it applies

You need a proper rational function, $\deg P < \deg Q$. If the numerator's degree is equal or higher, first do polynomial long division. The denominator $Q(x)$ must be factored into linear and/or irreducible quadratic factors.

The standard forms

For $\frac{P(x)}{Q(x)}$ with $Q$ fully factored, each factor contributes a term:

• Distinct linear $(ax+b)$ $\to$ $\dfrac{A}{ax+b}$.
• Repeated linear $(ax+b)^n$ $\to$ $\dfrac{A_1}{ax+b}+\dfrac{A_2}{(ax+b)^2}+\cdots+\dfrac{A_n}{(ax+b)^n}$.
• Irreducible quadratic $(ax^2+bx+c)$ $\to$ $\dfrac{Ax+B}{ax^2+bx+c}$ — a linear numerator, not just a constant.

The unknown constants are found from the resulting equations; the decomposition is unique, which is what lets us solve for them systematically.

Worked example

Decompose $\frac{3x+5}{x^2-3x+2}$. Factor the denominator: $(x-1)(x-2)$. Set

Clear denominators: $3x+5 = A(x-2)+B(x-1)$. Substituting the roots, $x=1$ gives $8=-A$ so $A=-8$, and $x=2$ gives $11=B$. Hence