Q.Integrate : $\displaystyle\int\dfrac{\operatorname{cosec}x}{\operatorname{cosec}x+\cot x}\,dx$.

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: