Question

14) Express the integrand as a sum of partial fractions and evaluate the integral

$\int \frac{x^{2}+4x+5}{(x-1)(x^{2}+4)}dx$

Answer 1

n² +4x+5 (n-1) (2+4) using Partial fraction decomposition +4215 Bxt 6 (2-1) (2244) - A & But 6 " x²+4) N-1 = x²+42 +5 (0-1) (22+4). A (2²74) + (Bat C) (0-1) CDC+4). n²+42t 1+A+5 10 - A(x+4) + (BrtC) (x-1). put n=1 = A(144) to - SA A= to 2 (A=2] compare the coff. of x² of now constant term. A+ B = 1 4A ~ (=5 -1 A=2 in eq. 1 Put 2+8 = 1

A=2 in eg. (11 put 442-(=5 (8-5 (C=3] Hence +4215 (2-)) (x²+4) x+3 2244 now a dat & go to an dla pt dn t f 7 - 12 de 2 dn 2-1) + 1 2 3 4 kom - 2 non = 0 &n/x-1) + I don'( 74). - data on Put n²+4= u anda = du = ce lola-1) + I don' ()- J elena • 2 dn(n-11 at & tant (A) - I inluste put value of 1. 10-1) + I tant (19- dn] %+41 +