Consider the rational function G(x)= 5x^ 2 +7x+2 (x-1) (x^ 2 +4x+5 )^ 2

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

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. (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. (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.

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