Question

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(\csc ^{2} \theta+2 \theta^{2}-3 \theta\right) d \theta$$

Short answer

Answer: The indefinite integral is \(-\cot \theta + \frac{2}{3} \theta^{3} - \frac{3}{2} \theta^{2} + C\).

Step-by-step solution

Integrate Each Term Separately

The first step is to split the integral into three separate integrals for each term: $$\int\left(\csc ^{2} \theta+2 \theta^{2}-3 \theta\right) d \theta = \int\csc ^{2} \theta d \theta + \int 2 \theta^{2} d \theta - \int 3 \theta d \theta$$

Integrate Cosecant Squared Function

Integrate the first function, the square of the cosecant function: $$\int\csc ^{2} \theta d \theta = -\cot \theta + C_1$$

Integrate Quadratic Term

Integrate the second function, the quadratic term: $$\int 2 \theta^{2} d \theta = \frac{2}{3} \theta^{3} + C_2$$

Integrate Linear Term

Integrate the third function, the linear term: $$\int 3 \theta d \theta = \frac{3}{2} \theta^{2} + C_3$$

Combine Integration Results

Now, add the results from Steps 2, 3, and 4 together: $$-\cot \theta + \frac{2}{3} \theta^{3} + C_1 - \left(\frac{3}{2} \theta^{2} + C_2\right) = -\cot \theta + \frac{2}{3} \theta^{3} - \frac{3}{2} \theta^{2} + (C_1 - C_2)$$ The constants \(C_1\) and \(C_2\) can be combined into a single constant, \(C\). The indefinite integral is then: $$-\cot \theta + \frac{2}{3} \theta^{3} - \frac{3}{2} \theta^{2} + C$$

Differentiate the Result to Check

Differentiate the indefinite integral with respect to \(\theta\): $$\frac{d}{d\theta}\left(-\cot \theta + \frac{2}{3} \theta^{3} - \frac{3}{2} \theta^{2} + C\right) = \csc^2 \theta + 2\theta^2 - 3\theta$$ Since the differentiation of the indefinite integral provided in Step 5 matches the given function, our work is correct.