Problem 22 - Essential Calculus

Question

Find the indefinite integral. $$ \int \frac{\csc ^{2} t}{\cot t} d t $$

Step-by-Step Solution

Answer

The indefinite integral is \( - 2 \cot t + C \)

Step 1: Rewrite the integral in terms of sine and cosine

Rewrite the integral as \( \int \frac{1}{\sin^2t \cdot \cos t} dt \). We can rewrite \( \csc^2t \) as \( \frac{1}{\sin^2t} \) and \( \cot t \) as \( \cos t \), thus the integral becomes \( \int \frac{1}{\sin^2t \cdot \cos t} dt \).

Step 2: Simplification

Split the integral as \( \int \frac{1}{\sin^2t \cdot \cos t} dt \) changing to \( \int \frac{\cos t}{\sin^2t} dt - \int \frac{1}{\sin^2t} dt \).

Step 3: Solve the integrals

The first integral is the derivative of cotangent, thus, \( - \cot t \). The second integral is equal to \( \csc^2 t \) which is \( -\cot t + C \). Therefore, the solution is sum of both namely, \( - 2 \cot t + C \).