Problem 61
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. $$\int\left(\frac{1}{x}-\frac{5}{x^{2}+1}\right) d x$$
Step-by-Step Solution
Verified Answer
\( \ln |x| - 5\arctan(x) + C \) is the most general antiderivative.
1Step 1: Recognize the Form of Each Term
The integral given is \( \int \left( \frac{1}{x} - \frac{5}{x^{2}+1} \right) dx \). Recognize that each term can be integrated separately. The first term, \( \frac{1}{x} \), is a standard integral form, while the second term, \( \frac{-5}{x^{2}+1} \), involves a form related to trigonometric integration.
2Step 2: Integrate the First Term
The integral of \( \frac{1}{x} \) is the natural logarithm. Therefore, \[ \int \frac{1}{x} \, dx = \ln |x| + C_1 \].
3Step 3: Integrate the Second Term
Recognizing \( \frac{-5}{x^{2}+1} \) leads us to the derivative of \( \arctan(x) \). Thus, \[ \int \frac{-5}{x^{2}+1} \, dx = -5 \arctan(x) + C_2 \].
4Step 4: Combine the Integrals
Combine the results from steps 2 and 3. The general antiderivative is \[ \ln |x| - 5\arctan(x) + C \], where \( C = C_1 + C_2 \) is a constant of integration.
5Step 5: Verification by Differentiation
Differentiate the solution \( \ln |x| - 5\arctan(x) + C \) to verify it returns to the original integrand. Differentiating yields \( \frac{1}{x} - \frac{5}{x^2+1} \), which matches the original integrand, confirming the solution is correct.
Key Concepts
AntiderivativeIndefinite IntegralTrigonometric Integration
Antiderivative
In calculus, finding an antiderivative is often one of the first tasks when solving integrals. An "antiderivative" is a function whose derivative yields the original function. It's like working backwards from differentiation. For example, if you know that the derivative of a function \(f(x)\) is \(g(x)\), then \(f(x)\) is the antiderivative of \(g(x)\). When solving problems involving antiderivatives, remember:
- The process involves reversing differentiation.
- You usually add a constant of integration \(C\), since differentiation of a constant becomes zero.
In the exercise, we found that the function \( \ln |x| - 5\arctan(x) \) is an antiderivative of \( \frac{1}{x} - \frac{5}{x^2+1} \). By differentiating it, we confirmed that it leads back to the original integrand, proving that the solution is correct.
- The process involves reversing differentiation.
- You usually add a constant of integration \(C\), since differentiation of a constant becomes zero.
In the exercise, we found that the function \( \ln |x| - 5\arctan(x) \) is an antiderivative of \( \frac{1}{x} - \frac{5}{x^2+1} \). By differentiating it, we confirmed that it leads back to the original integrand, proving that the solution is correct.
Indefinite Integral
The indefinite integral is a concept that stems from antiderivatives. An "indefinite integral" of a function represents a family of functions which serve as antiderivatives for that function. When you solve \( \int f(x) \, dx \), you are seeking a function whose derivative is \( f(x) \). Indefinite integrals have certain unique traits:
- Their representation uses the integral symbol \( \int \).
- They include a constant of integration \( C \) to account for the family of antiderivatives.
- The result is a general solution representing all possible antiderivatives.
For the exercise given, the indefinite integral \( \int \left( \frac{1}{x} - \frac{5}{x^{2}+1} \right) dx \) is computed as \( \ln |x| - 5\arctan(x) + C \). This result indicates all potential antiderivatives for the function \( \frac{1}{x} - \frac{5}{x^{2}+1} \).
- Their representation uses the integral symbol \( \int \).
- They include a constant of integration \( C \) to account for the family of antiderivatives.
- The result is a general solution representing all possible antiderivatives.
For the exercise given, the indefinite integral \( \int \left( \frac{1}{x} - \frac{5}{x^{2}+1} \right) dx \) is computed as \( \ln |x| - 5\arctan(x) + C \). This result indicates all potential antiderivatives for the function \( \frac{1}{x} - \frac{5}{x^{2}+1} \).
Trigonometric Integration
Trigonometric integration refers to integrating functions that involve trigonometric expressions. This method can involve recognizing parts of the integrand that resemble derivatives of trigonometric functions.
In our given problem, explaining the second term \( \frac{-5}{x^{2}+1} \) involves a trigonometric function because its integration leads to \( -5\arctan(x) \). This association comes from knowing that \( \int \frac{1}{x^2+1} \, dx = \arctan(x) + C \). From this, multiplying by \(-5\) gives \( \int \frac{-5}{x^{2}+1} \, dx = -5\arctan(x) + C \).
When you encounter integrals involving expressions like \( \frac{1}{x^2+1} \), consider their connection to inverse trigonometric functions like \( \arctan \). This strategy is central to correctly evaluating these integrals.
In our given problem, explaining the second term \( \frac{-5}{x^{2}+1} \) involves a trigonometric function because its integration leads to \( -5\arctan(x) \). This association comes from knowing that \( \int \frac{1}{x^2+1} \, dx = \arctan(x) + C \). From this, multiplying by \(-5\) gives \( \int \frac{-5}{x^{2}+1} \, dx = -5\arctan(x) + C \).
When you encounter integrals involving expressions like \( \frac{1}{x^2+1} \), consider their connection to inverse trigonometric functions like \( \arctan \). This strategy is central to correctly evaluating these integrals.
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