ตอบ:
#sqrt (1 + x ^ 2) -1 / 2ln (เอบีเอส (sqrt (1 + x ^ 2) +1)) + 1 / 2ln (เอบีเอส (sqrt (1 + x ^ 2) -1)) + C #
คำอธิบาย:
ใช้ # U ^ 2 = 1 + x ^ 2 #, # x = sqrt (U ^ 2-1) #
# 2u (du) / (DX) = 2x #, # DX = (Udu) / x #
#intsqrt (1 + x ^ 2) / xdx = int (usqrt (1 + x ^ 2)) / x ^ 2DU #
# INTU ^ 2 / (U ^ 2-1) du = int1 + 1 / (U ^ 2-1) du #
# 1 / (U ^ 2-1) = 1 / ((U + 1) (U-1)) = A / (U + 1) + B / (U-1) #
# 1 = A (U-1) + B (U + 1) #
# U = 1 #
# 1 = 2B #, # B = 2/1 #
# U = -1 #
# 1 = -2A #, # A = -1/2 #
# int1-1 / (2 (U + 1)) + 1 / (2 (U-1)) du = U-1 / 2ln (เอบีเอส (U + 1)) + 1 / 2ln (เอบีเอส (U-1)) + C #
วาง # U = sqrt (1 + x ^ 2) # กลับมาให้:
#sqrt (1 + x ^ 2) -1 / 2ln (เอบีเอส (sqrt (1 + x ^ 2) +1)) + 1 / 2ln (เอบีเอส (sqrt (1 + x ^ 2) -1)) + C #
