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.

Accepted Answer

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 $$ \frac{1}{a + b} 
eq \frac{1}{a} + \frac{1}{b} . To $$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.

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.

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Concetti chiave

Linearity of Integration

Properties of Rational Functions

Counterexamples in Integration