Hvordan løser du dette integral?

Svar:

#int ("d" x) / (x ^ 2-1) ^ 2 #

# = 1/4 (ln (x + 1) -ln (x-1) - (2x) / (x ^ 2-1)) + C #

Forklaring:

#int ("d" x) / (x ^ 2-1) ^ 2 #

# = int ("d" x) / ((x + 1) ^ 2 (x-1) ^ 2)

Lad os nu gøre de partielle fraktioner. Antag det

# 1 / ((x + 1) ^ 2 (x-1) ^ 2) = A / (x + 1) + B / (x + 1) ^ 2 + C / (x-1) + D / (x -1) ^ 2 #

for nogle konstanter # A, B, C, D #.

Derefter, # 1 = A (x + 1) (x-1) ^ 2 + B (x-1) ^ 2 + C (x + 1) ^ 2 (x-1) + D (x + 1) ^ 2 #

Udvid for at få

# 1 = (A + C) x ^ 3 + (B + C + D-A) x ^ 2 + (2D-2B-A-C) x + A + B-C + D #.

Equate koefficienter:

# {(A + C = 0), (B + C + D-A = 0), (2D-2B-A-C = 0), (A + B-C + D = 1):} #

Løsning giver # A = B = D = 1/4 # og # C = -1/4 #.

Således er vores oprindelige integral

# 1 (1 / (4 (x + 1)) + 1 / (4 (x + 1) ^ 2) -1 / (4 (x-1)) + 1 / (4 (x-1) ^ 2)) "d" x #

# = 1 / 4LN (x + 1) -1 / (4 (x + 1)) - 1 / 4LN (x-1) -1 / (4 (x-1)) + C #

Forenkle:

# = 1/4 (ln (x + 1) -ln (x-1) - (2x) / (x ^ 2-1)) + C #