UFOP - Universidade Federal de Ouro Preto

Apuntes

UFOP - Universidade Federal de Ouro Preto ICEB - Departamento de Matem ́ atica 15 a LISTA: INTEGRAIS - MTM 122 - 2025 1. Calcule as seguintes integrais usando o m ́etodo de integra ̧c ̃ao por partes: (a) Z xe x dx (b) Z x 2 sen(x)dx (c) Z xe x (1 + x) 2 dx (d) Z e −t cos(πt)dt (e) Z sen(ln(x))dx (f) Z arccos(2x)dx (g) Z 3 x cos(x)dx (h) Z x arctan(x)dx (i) Z sec 3 (x)dx (j) Z (x − 1)e −x dx (k) Z e 1 x x 3 dx (l) Z x 3 √ 1 − x 2 dx (m) Z x cossec 2 (x)dx (n) Z x sec(x) tan(x)dx (o) Z x 3 sen(5x)dx (p) Z x 4 cos(2x)dx (q) Z x 4 e x dx (r) Z (x 5 − x 3 + x)e −x dx 2. Calcule as seguintes integrais que envolvem potˆencias de fun ̧c ̃oes trigonom ́etricas: (a) Z sen 2 (x) cos 4 (x) dx (b) Z tan 5 (x) sec 3 (x)dx (c) Z sen 2 (x) cos 2 (x)dx (d) Z sen 5 (x) p cos(x) dx (e) Z sen(x) tan 2 (x) dx (f) Z cotan 2 (2x) + cotan 4 (2x)
dx (g) Z cos 4 (x) sen 6 (x) dx (h) Z sen 4 (ax)dx (i) Z sen 3 (y) cos 4 (y)dy 3. Calcule as seguintes integrais, usando substitui ̧c ̃ao trigonom ́etrica: (a) Z √ 16 − x 2 x 2 dx (b) Z 1 x 3 √ x 2 − 9 (c) Z 1 x 2 √ 5 − x 2 dx (d) Z 1 x 2 √ x 2 − 7 dx (e) Z 1 x √ 25 − x 2 dx (f) Z x 2 √ 2x − x 2 dx (g) Z (16 − 9x 2 ) 3 2 x 6 dx (h) Z 1 (4x − x 2 ) 3 2 dx (i) Z ...