Evaluate the integral: I = integral dx/(sin² x sqrt(cot

Evaluate the integral: $$ I = \int \frac{dx}{\sin^2 x\,\sqrt{\cot x}}. $$

Step 1 · Set up the substitution

Let:

$$ t = \cot x. $$

Then:

$$ dt = -\csc^2x\,dx = -\frac{1}{\sin^2x}\,dx \quad \Longrightarrow \quad \frac{dx}{\sin^2x} = -dt. $$

Step 2 · Rewrite the integral

The integrand $\dfrac{1}{\sin^2x\sqrt{\cot x}}\,dx$ is exactly $\dfrac{1}{\sqrt{\cot x}}\cdot\dfrac{dx}{\sin^2x}$, so substituting gives:

$$ I = \int \frac{1}{\sqrt{t}}\,(-dt) = -\int t^{-1/2}\,dt. $$

Step 3 · Apply the power rule

Using $\int t^n\,dt = \dfrac{t^{n+1}}{n+1}$ with $n=-\tfrac12$:

$$ -\int t^{-1/2}\,dt = -\frac{t^{1/2}}{1/2} + C = -2\sqrt{t} + C. $$

Step 4 · Substitute back

Substituting back $t=\cot x$:

$$ I = -2\sqrt{\cot x} + C. $$

Summary Table

Additional Notes

• The integrand requires $\cot x\geq0$ for the square root to be real, so $x$ is restricted to intervals such as $(0,\tfrac\pi2]$.
• A common pitfall here is to split $\sqrt{\cot x}=\sqrt{\cos x}/\sqrt{\sin x}$ and try to substitute $u=\sin x$ directly — this leaves a mismatched power of $\sin x$ that does not reduce cleanly. Recognizing that $1/\sin^2x$ is the derivative of $\cot x$ (up to sign) is the key that makes the whole integral a one-line power-rule problem.
• The result can be checked by differentiating $-2\sqrt{\cot x}$ with the chain rule: $\dfrac{d}{dx}\left[-2(\cot x)^{1/2}\right] = -2\cdot\tfrac12(\cot x)^{-1/2}\cdot(-\csc^2x) = \dfrac{\csc^2x}{\sqrt{\cot x}} = \dfrac{1}{\sin^2x\sqrt{\cot x}}$, which matches the original integrand.