Question

(5.6.21 from Stewart and Day) If a factor of the denominator of a rational function is an irreducible quadratic, such as x2+1, the corresponding partial fraction has a linear numerator. For instance, Bx+c f(x) = 2x2+x+1 Determine the values of A, B, and C and use them to evaluate x2+1 x(x2+1) f(x)dx. Use C as your constant of integration, and don't forget to use absolute value bars where needed. | axlox f(x) dx =

Answer 1

&. 5.6.21) $09) Join , A n 2 1 2 #B- = 2x²+x+1 Hla²+1) 2x²+x+1 Barc A(1²1) 712+) 20² ntl A+) +(Bxtc). X put x = 0 i put nai o to the A () to 20+1+1=1+1) + FA=! (B+C) 4= 2 + Btc. B + CO2 O- Similarly, put x = -1 26-1)?-X+X= alle 1)?+) + (-B+C)(-1) Solve & ③ we get B+ c = 2 + B-K = 0 2B B=14 & [c=1 f(x) = 2x2+XTI x (1²+1) I + (1xtel) 거 2121 ) fix) + x 4²+1 X x²+1

Now, ffcalda - f G 1 da 2 1²+1 + X²+1 dnt x e da x²+1 + (dx x²+1 X Lda + 1 je 2x X²+1 + da u²+1 findne= ln (1x1) + 5x les fix?+1) + fan' () + (3) 5-6:20) = +1)=52732-1 x2(x+2) Son 5x²+38-1 (x+2) A B c + x2(x+2) n 지고 5712734-1 x²(x+2) Ax+B X² + (arz) 5x²+32-1 Axi (x+2) + BCx+2) + C (23) put a=o. - B. o + B (2) to -V 2. Similarly, put x = -2 5(-2)² +31-2) - 1 = 0+0+ C (-2)² 20-6-1-4C

... 13=4C 13 4 pert x=1 56112 +361)-13 A (1)(3.) A(1) (3.) -1 (3) + 13 (1)2 २ 4 A 7 = 3A 3 +13 4 7=3A + 7 4 3A 7-1 BA 21 =) As I 4 4. Now Now, fooda wodne + 13 222 46x+2) Ilida Sh. 4 Zen(121) fax 2 22 + 13 (dx 4 (2+2) + 13 In (12+21) + 4. . 1 2། - ། 4 २ 7 ln (121) t / + 13 ln (12+24) +c. 4 4 2x

Now, I = -cos(80).é - Cos(80).20-81-Sin (80)ė ° +85] I - cos (80) ·ē +8 Sin (80) ē 0 - 64. I I +6.6 cos (80) + 8 Sin(so) 8 Sin(80). é o cos(80). és 65 I 2-8 da Scorci 6(1)(2-6) A x+7 + B X-6 G+1) 62-6) ..( 2-8) A (3-6) '+ (x+7) put a=6 O + B(13) 2 B - 2 13 put x = -7. 15 - Al-13) to A = 15 13

put y = -2 4.(-2) ² - 562) -12 B(-2)(-5) to IS Lot 16 B LOB 7 5 7 3 + + dy S ay 2 -sy-12 dy =( 5 (y +2) 564-3) S a day dy = 2 in (lyl) + 1 Inflytal) + 3 ln (14-31) tc + 1 3 to ل Yt2 y-3- 5 Apply limits S2 ln (lyl) + 1 in (ly+21) + 3 in (14-31)} 5 کی 2 10 (121) + 3 in (141) + 3 1n (1-11) en (111) + 7 6 (131) + 3 in (1-21) 2 ln (2) + 2 ln (4) to - 240-I ln (3) - en (2) Z ln (2) + 2 m (4) - Zln (3) 1:37 316T کی 3 5 کر 5 22