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UNIVERSIDADE FEDERAL DE CAMPINA GRANDE UNIDADE ACADEˆMICA DE MATEMA´TICA Disciplina: Ca´lculo Diferencial e Integral I Per´ıodo 2014.1 Professora: Itailma Rocha 5a Lista de Exerc´ıcios - Gabarito 1 - . (a) 7 2 (b) 2 (c) 0 (d) 4 9 (e) 10 (f) 3 4 (g) −1 (h) 16 3 (i) 2 (j) 12 (k) 39 32 (l) 19 3 (m) ln 2 + 9 2 (n) −2 3 (o) 2−√2 2 (p) 1 2 (e2 − 1) (q) pi (r) pi 4 (s) pi 4 (t) 1 2 - . (a) 20 (b) 14 3 (c) 4 3 (d) 32 3 (e) 4 (f) 1 (g) 23 3 (h) 21 (i) 2 (j) 1 4 (k) 1 2 (l) 7 3 (m) 2( √ 2− 1) (n) 1 6 (o) 9 2 (p) 4 3 1 3 - . (a) (3x− 2)4 12 + k (b) 2 9 √ (3x− 2)3 + k (c) − 1 3(3x− 2)2 + k (d) 1 3 ln |3x− 2|+ k (e) −1 2 cosx2 + k (f) 1 2 ex 2 + k (g) 1 3 ex 3 + k (h) 1 4 sen(x4) + k (i) −cos 4 x 4 + k (j) sen6x 6 + k (k) 2 ln |x+ 3|+ k (l) 5 4 ln |4x+ 3|+ k (m) 1 8 ln(1 + 4x2) + k (n) 1 4 ln(5 + 6x2) + k (o) − 1 8(1 + 4x2) + k (p) 1 9 √ (1 + 3x2)3 + k (q) 2 3 √ (1 + ex)3 + k (r) 1 cosx + k 4 - . (a) sen3x 3 + k (b) sen3x 3 − sen 5x 5 + k (c) − cosx+ 1 3 cos3 x+ k (d) senx− 2sen 3x 3 + sen5x 5 + k (e) 1 4 tg4x+ k (f) tg2x 2 + k (g) sec3 x 3 + k (h) sec6 x 6 − sec 4 x 4 + k 2 (i) 1 2 cos2 x + k (j) sec x+ cosx+ k (k) 1 2 ln |3 + 2tgx|+ k (l) sen 4x 4 − sen 6x 6 + k 5 - . (a) (x− 1)ex + k (b) −xcosx+ senx+ k (c) ex(x2 − 2x+ 2) + k (d) x2 2 (lnx− 1 2 ) + k (e) x(lnx− 1) + k (f) x3 3 (lnx− 1 3 ) + k (g) xtgx+ ln | cosx|+ k (h) x2 2 (ln2 x− lnx+ 1 2 ) + k (i) x ln2 x− 2x(lnx− 1) + k (j) e2x 2 (x− 1 2 ) + k (k) ex 2 (senx+ cosx) + k (l) −e −2x 5 (cosx+ 2senx) + k (m) 1 2 (x2 − 1)ex2 + k (n) 1 2 (x2senx2 + cosx2) + k (o) e−x 5 (2sen2x− cos 2x) + k (p) x 2 cosx+ 2xsenx+ 2 cosx+ k 6 - . (a) 1 (b) 2 ln 2− 1 (c) epi/2 − 1 2 (d) −x2e−sx 2 − 2xe −st s2 + 2e−sx s3 + 2 s3 (e) 8 √ 2− 2 15 (f) e− 1 3 (g) −1 2 (h) ln 9 3 (i) 2 ln 2− 3 4 (j) −2pi (k) ln 5− ln 3 (l) 4 ln 2− 2 (m) −2 (n) 3 (o) 36 (p) 55 63 (q) pi 4 + 1 (r) 1− ln 4 (s) 1 11 − 9 ln 10 (r) 5 2 + 4 ln 5 (u) pi 6 (v) e−√e (w) 2 (v) 16 15 7 - . (a) (3x− 2)21 63 + k (b) 1 3 3 √ (2x+ x2)2 + k (c) 1 1− eu + k (d) ln3 x 3 + k (e) −2 3 cos(1 + x3/2) + k (f) − cos(5x) ln 5 + k (g) ln |arcsenx|+ k (h) etgx + k (i) arctgx+ 1 2 ln |1 + x2|+ k (j) (2x+ 5) 10 40 − 5 36 (2x+ 5)9 + k (k) 1 2 arctg(x2) + k (l) ln(2x + 3) ln 2 + k 4 (m) (x3 + 3x)5 15 + k (n) arctg2x 2 + k (o) − cos(lnx) + k (l) 2√1 + tgx+ k 8 - . (a) 1 4 ln |x− 2| − 1 4 ln |x+ 2|+ k (b) 3 ln |x− 3| − 2 ln |x− 2|+ k (c) 1 2 ln |x− 2|+ 1 2 ln |x+ 2|+ k (d) ln |x− 1| − 4 x− 1 + k (e) x+ 19 4 ln |x− 3|+ 1 4 ln |x+ 1|+ k (f) x 3 3 + 2x+ 4 ln |x− 1|+ 3 x− 1 + k (g) x+ 2 ln |x− 3| − 2 ln |x+ 3|+ k (h) 1 3 ln |x− 2| − 1 3 ln |x+ 1|+ k 9 - . (a) −1 2 ln |x|+ 3 10 ln |x− 2| − 2 15 ln |x+ 3|+ k. (b) x2 2 − ln |x|+ 3 2 ln |x− 1|+ 1 2 ln |x+ 1|+ k. (c) 2 9 ln |x+ 2| − 2 9 ln |x− 1|+ 2 3(x− 1) + k. (d) −2 ln |x− 1|+ 5 3 ln |x− 2|+ 1 3 ln |x+ 1|+ k. (e) x3 3 + 4x+ 35 8 ln |x− 2|+ 29 8 ln |x+ 2| − 3 4 ln |x|+ k. (f) x+ 3 2 ln |x− 2| − 1 2 ln |x|+ k. (g) 2 ln |x− 1|+ ln |x2 + 6x+ 10|+ arctg(x+ 3) + k. (h) 2 5 ln |x| − 1 5 ln |x2 + 2x+ 5|+ 3 10 arctg( x+ 3 2 ) + k. (i) 2 ln |x2 + 6x+ 12|+ 11√ 3 arctg( x+ 3√ 3 ) + k. (j) 2 ln |x− 1|+ 1 2 ln |x2 + 2x+ 3|+ √ 2 2 arctg( x+ 1√ 2 ) + k. 5 (k) x+ 2 ln |x− 1|+ 1 2 ln |x2 + 2x+ 3|+ √ 2 2 arctg( x+ 1√ 2 ) + k. (l) ln |x− 2|+ 1 2 ln |x2 + 2x+ 4| − √ 3 3 arctg( x+ 1√ 3 ) + k. 10 - (a) 25 2 arcsen( x 2 )− x √ 25− x2 2 + k (b) −√25− x2 + k (c) x 2 √ 4 + x2 − ln | √ 4 + x2 + x 2 | (d) x √ 4 + 9x2 18 − 2 7 ln | √ 4 + 9x2 + 3x 2 |+ k (e) (x2 − 32)√x2 − 16 3 (f) √ x2 − 4− 2arcsec(x 2 ) + k 6
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