(i) Show that x^ 4 +2x^ 3 +3x^ 2 +4x+5 (x^ 2 +1)^ 2 can be written as a sum of a…

MATHEMATICS · Subjective · CBSE · Mathematics · Integrals · Integrals

Problem

(i) Show that

\frac{x^{4}+2x^{3}+3x^{2}+4x+5}{(x^{2}+1)^{2}}

can be written as a sum of a polynomial and a rational function whose denominator is

(x^{2}+1)^{2}

. (ii) Further resolve the rational part obtained in (i) into partial fractions of the form

\frac{Ax+B}{x^{2}+1}+\frac{Cx+D}{(x^{2}+1)^{2}}.

(iii) Hence compute the integral

\int \frac{x^{4}+2x^{3}+3x^{2}+4x+5}{(x^{2}+1)^{2}}\,dx.