Mathematics

Mathematics

Problem

Consider the rational function G(x)=5x2+7x+2(x−1)(x2+4x+5)2.G(x)=\frac{5x^{2}+7x+2}{(x-1)\bigl(x^{2}+4x+5\bigr)^{2}}. Answer the following. a. (i) Rewrite G(x)G(x) as a sum of terms having denominators (x−1)(x-1), (x2+4x+5)(x^{2}+4x+5) and (x2+4x+5)2(x^{2}+4x+5)^{2}, each with a linear numerator where appropriate. b. (ii) Using the result of part (i), evaluate the indefinite integral ∫G(x) dx\displaystyle\int G(x)\,dx, presenting your answer in its simplest form.
Intuitive explanation

The key to decomposing this rational function is partial fraction decomposition, which breaks a complex fraction into simpler pieces you can integrate or analyze. Notice the denominator has a linear factor (x−1)(x-1) and a repeated irreducible quadratic (x2+4x+5)2(x^2+4x+5)^2—this means you'll need separate terms for each, including one for the repeated quadratic's second power. Set up the decomposition with unknown coefficients, then solve by clearing denominators and matching powers of xx.

Intuitive explanation

The key to decomposing this rational function is partial fraction decomposition, which breaks a complex fraction into simpler pieces you can integrate or analyze. Notice the denominator has a linear factor (x−1)(x-1) and a repeated irreducible quadratic (x2+4x+5)2(x^2+4x+5)^2—this means you'll need separate terms for each, including one for the repeated quadratic's second power. Set up the decomposition with unknown coefficients, then solve by clearing denominators and matching powers of xx.

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