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Chapter 4: Problem 105

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

Expert verified
Question: Find the indefinite integral of the given function: $$\int\left(\csc ^{2} \theta+2 \theta^{2}-3 \theta\right) d \theta$$ Answer: The indefinite integral of the given function is $$\int\left(\csc ^{2} \theta+2 \theta^{2}-3 \theta\right) d \theta = - \cot \theta + \frac{2}{3}\theta^3 - \frac{3}{2}\theta^2 + C$$ where C is the general constant of integration.

Step by step solution

01

Understand the function and split the integral into separate terms

In this problem, we will find the indefinite integral of the given function. The function has three terms, and we can split the integral into separate terms for simplification: $$\int\left(\csc ^{2} \theta + 2\theta^{2} - 3\theta\right) d\theta = \int \csc^2 \theta d\theta + \int 2\theta^2 d\theta - \int 3\theta d\theta$$
02

Integrate the first term

To integrate the first term, recall the trigonometric identity: $$\int \csc^2 \theta d\theta = - \cot \theta + C_1$$ where \(C_1\) is the constant of integration.
03

Integrate the second term

To integrate the second term, apply the power rule for integration: $$\int 2\theta^2 d\theta = \frac{2}{3}\theta^3 + C_2$$ where \(C_2\) is the constant of integration.
04

Integrate the third term

To integrate the third term, apply the power rule for integration: $$\int 3\theta d\theta = \frac{3}{2}\theta^2 + C_3$$ where \(C_3\) is the constant of integration.
05

Combine the terms

Now that we have found the integral of each term, we can combine them to find the integral of the entire function: $$\int\left(\csc ^{2} \theta+2 \theta^{2}-3 \theta\right) d \theta = - \cot \theta + \frac{2}{3}\theta^3 - \frac{3}{2}\theta^2 + C$$ where \(C = C_1+C_2+C_3\) is the general constant of integration.
06

Verify the result by differentiation

To verify that this is the correct indefinite integral, we must differentiate the result and obtain the original function: $$\frac{d}{d\theta}\left(- \cot \theta + \frac{2}{3}\theta^3 - \frac{3}{2}\theta^2 + C\right) = -(-\csc^2 \theta) + 2\theta^2 - 3\theta = \csc^2 \theta + 2\theta^2 - 3\theta$$ This matches the original function, confirming that our solution is correct.

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