Integratie met substitutie intsqrt (1 + x ^ 2) / x dx? Hoe los ik deze vraag op, help me alstublieft?

Antwoord:

#sqrt (1 + x ^ 2) -1 / 2ln (abs (sqrt (1 + x ^ 2) 1)) + 1 / 2ln (abs (sqrt (1 + x ^ 2) -1)) + C #

Uitleg:

Gebruik # 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 = 1/2 #

# U = -1 #

# 1 = -2A #, # A = -1/2 #

# Int1-1 / (2 (u + 1)) + 1 / (2 (u-1)) du = u-1 / 2ln (abs (u + 1)) + 1 / 2ln (abs (u-1)) + C #

putting # U = sqrt (1 + x ^ 2) # terug in geeft:

#sqrt (1 + x ^ 2) -1 / 2ln (abs (sqrt (1 + x ^ 2) 1)) + 1 / 2ln (abs (sqrt (1 + x ^ 2) -1)) + C #