Problem 39
Question
In Exercises \(17-56,\) 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. $$\int\left(-3 \csc ^{2} x\right) d x$$
Step-by-Step Solution
Verified Answer
The indefinite integral is: \( 3 \cot x + C \).
1Step 1: Identify the integral form
Before solving the integral, identify the standard integral form you are dealing with. The integral \( \int -3 \csc^2 x \, dx \) is a constant multiple of \( \int \csc^2 x \, dx \).
2Step 2: Recall standard integral result
Recall that the antiderivative of \( \csc^2 x \) with respect to \( x \) is \(-\cot x\). This involves understanding trigonometric integrals.
3Step 3: Apply the constant multiple property
Now, apply the constant multiple property of integrals: \( \int -3 \csc^2 x \, dx = -3 \int \csc^2 x \, dx \). This allows us to factor the constant out of the integral.
4Step 4: Compute the integral
Integrating \( \csc^2 x \) gives \(-\cot x\). Therefore, the integral is: \(-3(-\cot x) = 3 \cot x + C\), where \( C \) is the constant of integration.
5Step 5: Verify by differentiation
Differentiate \( 3 \cot x + C \) to ensure that it matches the original integrand. Differentiating \( 3 \cot x \) gives \(-3 \csc^2 x\), which is indeed the original expression inside the integral. Thus, verification confirms our solution.
Key Concepts
Indefinite IntegralsTrigonometric IntegralsIntegration Verification
Indefinite Integrals
Indefinite integrals are a fundamental concept in calculus. They represent a broad family of functions without specified boundaries. An indefinite integral, also known as an antiderivative, essentially reverses the process of differentiation. When you calculate an indefinite integral, you're finding a function whose derivative yields the given integrand.
The notation for indefinite integrals is written with the integral sign \( \int \), followed by the function \( f(x) \), and the differential \( dx \). This signals the integration of \( f(x) \) with respect to \( x \). The result is expressed in the form \( F(x) + C \), where \( F(x) \) is the antiderivative and \( C \) is an arbitrary constant of integration. This constant is crucial, as the derivative of a constant is zero, allowing a family of solutions differing by a constant value.
The notation for indefinite integrals is written with the integral sign \( \int \), followed by the function \( f(x) \), and the differential \( dx \). This signals the integration of \( f(x) \) with respect to \( x \). The result is expressed in the form \( F(x) + C \), where \( F(x) \) is the antiderivative and \( C \) is an arbitrary constant of integration. This constant is crucial, as the derivative of a constant is zero, allowing a family of solutions differing by a constant value.
- When you find the indefinite integral of \( f(x) \), formulate \( F(x) + C \).
- The derivative of \( F(x) + C \) should return the original function \( f(x) \).
- This process is essential for solving differential equations and in various fields of applied mathematics.
Trigonometric Integrals
Trigonometric integrals involve the integration of trigonometric functions such as sine, cosine, tangent, and their respective reciprocals like cosecant, secant, and cotangent. These are common in calculus and help solve problems involving periodic functions.
For example, integrating \( \csc^2 x \) is a frequent problem you'll encounter. Knowing the antiderivatives of basic trigonometric functions is essential. Specifically, the antiderivative of \( \csc^2 x \) is \(-\cot x\).
For example, integrating \( \csc^2 x \) is a frequent problem you'll encounter. Knowing the antiderivatives of basic trigonometric functions is essential. Specifically, the antiderivative of \( \csc^2 x \) is \(-\cot x\).
- Recognize and remember common integral forms, such as \( \int \csc^2 x \, dx = -\cot x + C \).
- Use symmetry and identities to simplify integrals and facilitate integration.
- Blending this knowledge with algebraic techniques like substitution can make complex integrals more manageable.
Integration Verification
Integration verification is a critical step to ensure the accuracy of your integration. After calculating an integral, it's important to verify that the antiderivative is correct by differentiating your result. This step helps confirm that the antiderivative solution returns the original integrand.
In our example, the process involved integrating \(-3 \csc^2 x\) to obtain \(3 \cot x + C\). To verify, differentiate \(3 \cot x + C\). The derivative of \(\cot x\) is \(-\csc^2 x\), so differentiating \(3 \cot x\) yields \(-3 \csc^2 x\), which is indeed the original function we integrated.
In our example, the process involved integrating \(-3 \csc^2 x\) to obtain \(3 \cot x + C\). To verify, differentiate \(3 \cot x + C\). The derivative of \(\cot x\) is \(-\csc^2 x\), so differentiating \(3 \cot x\) yields \(-3 \csc^2 x\), which is indeed the original function we integrated.
- Always differentiate your antiderivative to check the solution, ensuring you retrieve the original function.
- This not only verifies accuracy but also solidifies understanding.
- Developing this habit prevents errors and fosters confidence in solving integrals, especially when dealing with more complex functions.
Other exercises in this chapter
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