Problem 67
Question
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\begin{aligned} &\int \cot ^{2} x d x\\\ &\text { (Hint: }\left.1+\cot ^{2} x=\csc ^{2} x\right) \end{aligned}$$
Step-by-Step Solution
Verified Answer
The most general antiderivative is \(-\cot x + x + C\).
1Step 1: Use the Trigonometric Identity
To solve this problem, we will use the hint provided: \(1 + \cot^2 x = \csc^2 x\). This identity allows us to rewrite the integrand. We have \(\cot^2 x = \csc^2 x - 1\). Thus, the integral becomes: \[ \int \cot^2 x \, dx = \int (\csc^2 x - 1) \, dx. \]
2Step 2: Split the Integral
Now we will split the integral into two separate integrals: \[ \int (\csc^2 x - 1) \, dx = \int \csc^2 x \, dx - \int 1 \, dx. \] This separation makes it easier to tackle each part individually.
3Step 3: Integrate Each Part
For the first integral: \( \int \csc^2 x \, dx = -\cot x + C_1 \), as the derivative of \(-\cot x\) is \( \csc^2 x\). For the second integral: \( \int 1 \, dx = x + C_2 \), as the integration of \(1\) with respect to \(x\) is \(x\). Thus, the integral becomes: \[ -\cot x + x + C. \] Where \(C = C_1 - C_2\) is the constant of integration.
4Step 4: Verify by Differentiation
To ensure the solution is correct, differentiate the result with respect to \(x\):\[\frac{d}{dx}(-\cot x + x + C) = \frac{d}{dx}(-\cot x) + \frac{d}{dx}x = \csc^2 x - 1. \]This equals to \(\cot^2 x\), confirming that the integration was performed correctly.
Key Concepts
Trigonometric IdentitiesAntiderivativesIntegration Techniques
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable involved. These identities are crucial in calculus, particularly when simplifying complex expressions or calculating integrals. In our example, the identity \(1 + \cot^2 x = \csc^2 x\) is used to rewrite the integrand. Utilizing such identities is common when integrating trigonometric functions because they simplify the expressions, making it easier to find their antiderivatives.
By applying this identity, the integral of \(\cot^2 x\) can be simplified to \(\int (\csc^2 x - 1) \, dx\). Being familiar with key trigonometric identities allows us to seamlessly transform and manage these expressions during integration. Keep in mind other common identities like \(\sin^2 x + \cos^2 x = 1\) and \(1 + \tan^2 x = \sec^2 x\), as these often appear in integration problems.
By applying this identity, the integral of \(\cot^2 x\) can be simplified to \(\int (\csc^2 x - 1) \, dx\). Being familiar with key trigonometric identities allows us to seamlessly transform and manage these expressions during integration. Keep in mind other common identities like \(\sin^2 x + \cos^2 x = 1\) and \(1 + \tan^2 x = \sec^2 x\), as these often appear in integration problems.
Antiderivatives
Antiderivatives, or indefinite integrals, represent a family of functions that differ by a constant. They are the opposite of derivatives and help in finding the original function from its rate of change. In the problem provided, finding the antiderivative of \(\cot^2 x\) involves using trigonometric identities and integration techniques.
Specifically, after using the identity \(\cot^2 x = \csc^2 x - 1\), the problem is divided into simpler parts: \(\int \csc^2 x \, dx\) and \(\int 1 \, dx\). These separate integrals give antiderivatives \(-\cot x\) and \(x\), respectively. By adding these results and including the constant of integration \(C\), we achieve the most general form of the antiderivative: \(-\cot x + x + C\).
Verifying through differentiation is essential to confirm the correctness of the antiderivative found.
Specifically, after using the identity \(\cot^2 x = \csc^2 x - 1\), the problem is divided into simpler parts: \(\int \csc^2 x \, dx\) and \(\int 1 \, dx\). These separate integrals give antiderivatives \(-\cot x\) and \(x\), respectively. By adding these results and including the constant of integration \(C\), we achieve the most general form of the antiderivative: \(-\cot x + x + C\).
Verifying through differentiation is essential to confirm the correctness of the antiderivative found.
Integration Techniques
Integration techniques are various methods used to evaluate integrals, especially when faced with complex expressions. For trigonometric functions, these techniques often include using identities or algebraic manipulation to simplify integrals.
One integral in the problem is \(\int \csc^2 x \, dx\), which has a known antiderivative of \(-\cot x\). Such known results help streamline the integration process. The other integral, \(\int 1 \, dx\), is straightforward, resulting in \(x\).
Utilizing these techniques effectively requires practice and a good understanding of how different functions relate to each other. Remember, integration is about reversing differentiation, so knowing derivative rules can also guide you in finding integrals more efficiently.
One integral in the problem is \(\int \csc^2 x \, dx\), which has a known antiderivative of \(-\cot x\). Such known results help streamline the integration process. The other integral, \(\int 1 \, dx\), is straightforward, resulting in \(x\).
Utilizing these techniques effectively requires practice and a good understanding of how different functions relate to each other. Remember, integration is about reversing differentiation, so knowing derivative rules can also guide you in finding integrals more efficiently.
Other exercises in this chapter
Problem 67
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