Question

73. Two methods Evaluate ∫ dx/(x² - 1), for x \u003e 1, in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.

Accepted Answer

First, recognize that the integral is $$ \int \frac{dx}{x^2 - 1} $$ with $$ x \u003e 1 . $$Notice that the denominator factors as $$ x^2 - 1 = (x - 1)(x + 1) . $$This suggests using partial fractions to decompose the integrand. Set up the partial fraction decomposition: $$ \frac{1}{x^2 - 1} = \frac{A}{x - 1} + \frac{B}{x + 1} . $$Multiply both sides by $$ (x - 1)(x + 1)  to $$get $$ 1 = A(x + 1) + B(x - 1) . $$Solve for constants $$ A $$ and $$ B  by $$choosing convenient values for $$ x  or by $$equating coefficients. Once $$ A $$ and $$ B $$ are found, rewrite the integral as $$ \int \left( \frac{A}{x - 1} + \frac{B}{x + 1} ight) dx . $$Integrate each term separately to get logarithmic expressions involving $$ x - 1 $$ and $$ x + 1 . $$For the trigonometric substitution method, note that $$ x^2 - 1 $$ resembles $$ \sec^2 	heta - 1 = 	an^2 	heta . $$Use the substitution $$ x = \sec 	heta $$, which implies $$ dx = \sec 	heta 	an 	heta \, d	heta . $$Rewrite the integral in terms of $$ 	heta $$ and simplify. After substitution, the integral becomes $$ \int \frac{\sec 	heta 	an 	heta \, d	heta}{\sec^2 	heta - 1} = \int \frac{\sec 	heta 	an 	heta}{	an^2 	heta} d	heta = \int \frac{\sec 	heta}{	an 	heta} d	heta . $$Simplify and integrate with respect to $$ 	heta $$, then back-substitute $$ 	heta = \sec^{-1} x  to $$express the answer in terms of $$ x . $$Finally, compare this result with the one from partial fractions to reconcile the two expressions.

73. Two methods Evaluate ∫ dx/(x² - 1), for x > 1, in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.

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주요 개념

Partial Fraction Decomposition

Trigonometric Substitution

Reconciliation of Different Integral Forms