Textbook Question
Variations on the substitution method Evaluate the following integrals.
∫ (𝒵 + 1) √(3𝒵 + 2) d𝒵
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7. Antiderivatives & Indefinite Integrals
Indefinite Integrals
2:18 minutesProblem 8.1.65a
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. ∫(3/(x² + 4)) dx = ∫(3/x²) dx + ∫(3/4) dx.
Verified step by step guidance1
Recall the property of integrals that states \( \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \). This means the integral of a sum is the sum of the integrals, but this property does not apply to quotients or products inside the integral unless they are separated as sums.
Look carefully at the given integral: \( \int \frac{3}{x^2 + 4} \, dx \). The denominator is \( x^2 + 4 \), which is a sum inside the denominator, not a sum of two separate fractions.
The statement claims that \( \int \frac{3}{x^2 + 4} \, dx = \int \frac{3}{x^2} \, dx + \int \frac{3}{4} \, dx \). This would mean splitting the fraction \( \frac{3}{x^2 + 4} \) into \( \frac{3}{x^2} + \frac{3}{4} \), which is not algebraically correct because \( \frac{1}{a + b} \neq \frac{1}{a} + \frac{1}{b} \).
To verify, consider a counterexample by choosing a specific value of \( x \), for example \( x = 1 \), and compare \( \frac{3}{1^2 + 4} = \frac{3}{5} \) with \( \frac{3}{1^2} + \frac{3}{4} = 3 + \frac{3}{4} = \frac{15}{4} \). Since these are not equal, the equality inside the integral does not hold.
Therefore, the original statement is false because the integral of a quotient with a sum in the denominator cannot be split into the sum of integrals of separate fractions as claimed.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Linearity of Integration
Integration is a linear operation, meaning the integral of a sum is the sum of the integrals. However, this property applies only when the integrand is expressed as a sum of separate terms, not when terms are combined inside a single denominator.
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Linearization
Properties of Rational Functions
A rational function is a ratio of polynomials. Splitting an integral like ∫(3/(x² + 4)) dx into ∫(3/x²) dx + ∫(3/4) dx is incorrect because (x² + 4) is a single denominator, not a sum of separate fractions.
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Counterexamples in Integration
To test if an integral equality holds, substituting specific values or comparing derivatives can provide counterexamples. Here, evaluating or differentiating the proposed split shows it does not equal the original integral, disproving the statement.
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