Question

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.∫ (8x² + 8x + 2) / (4x² + 1)² dx

Accepted Answer

Recognize that the integrand is a rational function where the denominator is a repeated quadratic factor: $$(4x^{2}$$ + $$1)^{2}. $$Since the denominator is a repeated irreducible quadratic, set up the partial fraction decomposition accordingly. Express the integrand as a sum of partial fractions of the form: $$\frac{8x$$^{2}$$ + 8x + 2}{(4x$$^{2}$$ + 1$$)^{2}$$} = \frac{Ax + B}{4x$$^{2}$$ + 1} + \frac{Cx + D}{(4x$$^{2}$$ + 1$$)^{2}$$}$$, where $$A$$, $$B$$, $$C$$, and $$D$$ are constants to be determined. Multiply both sides of the equation by $$(4x^{2}$$ + $$1)^{2} to $$clear the denominators, resulting in: $$8x^{2} + 8x + 2 = (Ax$$ + $$B)(4x^{2} + 1) + (Cx + D$$)$$. Expand the right-hand side and collect like terms in powers of x$. Then, equate the coefficients of corresponding powers of x$ on both sides to form a system of equations for A$, B$, C$, and D$. Solve the system of equations to find the values of A$, B$, C$, and D$. Once found, rewrite the integral as the sum of simpler integrals involving these partial fractions, which can then be integrated using standard techniques such as substitution and recognizing derivatives of inverse trigonometric functions.

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (8x² + 8x + 2) / (4x² + 1)² dx

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Key Concepts

Partial Fraction Decomposition

Integration of Rational Functions

Handling Repeated Quadratic Factors