Apéndice B Integrales B.1 Fórmulas más usadas ˆ ˆ un du = un+1 + c; n �= −1 n+1 n+1 n (au + b) du = (B.3) 1 1 du = ln (au + b) + c au + b a (B.4) ˆ ˆ ˆ ueau du = ˆ 1 au e +c a (B.5) 1 au e (au − 1) + c a2 (B.6) eau du = un eau du = ˆ (B.2) 1 du = lnu + c u ˆ ˆ (au + b) + c; n �= −1 a (n + 1) (B.1) n un eau − a a ˆ un−1 eau du (B.7) ln udu = u ln u − u (B.8) 1 sen (au) du = − cos (au) + c a (B.9) 391 392 APÉNDICE B. INTEGRALES ˆ B.2 cos (au) du = 1 sen (au) + c a (B.10) ˆ 1 tg (au) du = − ln cos (au) + c a (B.11) ˆ ctg (au) du = 1 ln sen (au) + c a (B.12) 1 eau [b sen (bu) + a cos (bu)] + c + b2 (B.13) 1 eau [a sen (bu) − b cos (bu)] + c a 2 + b2 (B.14) ˆ eau cos (bu) du = ˆ eau sen (bu) du = a2 Ortogonalidad y paralelismo ˆ2π ˆ2π cos (mx) cos (nx) dx = sen (mx) sen (nx) dx = ˆ2π ˆ2π π; 0; π; 0; m=n (B.15) m �= n m=n (B.16) m �= n sen (mx) cos (nx) dx = 0 (B.17) 2π ; (B.18) cos (nx) dx = ˆ2π 0; n=0 n = 1, 2, . . . sen (nx) dx = 0 (B.19) B.2. ORTOGONALIDAD Y PARALELISMO ˆ2π ˆ2π ˆ2π ˆ2π 393 cos2 (nx) dx = π (B.20) sen2 (nx) dx = π (B.21) ejkx dx = 2π; 0; ejmx e−jnx dx = k=0 (B.22) k = 1, 2, · · · 2π; 0; m=n (B.23) m �= n 394 APÉNDICE B. INTEGRALES