Question

7–84. Evaluate the following integrals.16. ∫ [1 / (x⁴ – 1)] dx

Accepted Answer

Start by recognizing that the integrand is a rational function: $$\frac{1}{x$$^{4}$$ - 1}. $$The first step is to factor the denominator $$x^{4} - 1$ using the difference of squares formula: a$$^{2}$$ - b$$^{2}$$ = (a - b)(a + b). $$Here, $$x^{4} - 1$$ = $$(x^{2}$$)^{2}$$ - 1$$^{2}$$ = (x$$^{2}$$ - 1)(x$$^{2}$$ + 1). $$Next, factor $$x^{2} - 1$ further since it is also a difference of squares: x$$^{2}$$ - 1 = (x - 1)(x + 1). So $$the full factorization of the denominator is $$x^{4} - 1 = (x - 1)(x$$ + $$1)(x^{2} + 1$$)$$. Rewrite the integral as $$\int \frac{1}{(x - 1)(x + $$1)(x^{2} + 1$$)} \, dx$$. The next step is to decompose the integrand into partial fractions of the form: $$\frac{A}{x - 1} + \frac{B}{x + 1} + \frac{Cx + D}{$$x^{2} + 1$$}$$, where A$, B$, C$, and D$ are constants to be determined. Multiply both sides of the equation by the denominator (x - 1)(x$$ + $$1)(x^{2} + 1$$)$$ to clear the fractions, resulting in an identity involving polynomials. Then, equate the coefficients of corresponding powers of x$ on both sides to form a system of linear equations for A$, B$, C$, and D$. Solve the system of equations to find the values of A$, B$, C$, and D$. Once these constants are found, rewrite the integral as a sum of simpler integrals: $$\int \frac{A}{x - 1} \, dx + \int \frac{B}{x + 1} \, dx + \int \frac{Cx + D}{$$x^{2} + 1$$} \, dx$. Each of these integrals can then be evaluated using standard integration techniques.

7–84. Evaluate the following integrals.
16. ∫ [1 / (x⁴ – 1)] dx

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Key Concepts

Partial Fraction Decomposition

Factoring Polynomials

Integration of Rational Functions