Which of the following methods is most efficient for evaluating the integral: int cot4(x)csc^2(x)dx Select only ONE answer. straight-forward u-substitution Dusing trig integration (with the identity csc^2(x)=cot^2(x)-1) Dre-write everything in terms of sine and cosine using trig integration (by first splitting up the integrand as cot^3(x)csc(x)cot(x)csc(x) then using the appropriate trig identity

## Step 1: Analyze the Integral<br />### The integral $\int \cot^4(x)\csc^2(x) dx$ has a $\csc^2(x)$ term, which is the derivative of $-\cot(x)$. This suggests a u-substitution with $u = \cot(x)$.<br /><br />## Step 2: Perform u-substitution<br />### Let $u = \cot(x)$. Then $du = -\csc^2(x) dx$. Substituting into the integral, we get $\int u^4 (-du) = -\int u^4 du$.<br /><br />## Step 3: Evaluate the integral<br />### $-\int u^4 du = -\frac{u^5}{5} + C$.<br /><br />## Step 4: Substitute back<br />### Substituting back $u = \cot(x)$, we get $-\frac{\cot^5(x)}{5} + C$.<br /><br />## Step 5: Efficiency Comparison<br />### The u-substitution method is the most efficient as it directly addresses the integral's form and avoids more complex trigonometric manipulations. Rewriting in terms of sine and cosine or using other trigonometric identities would lead to more complicated expressions and calculations.