Ako sa integrujete? 1 / (x ^ 2 + 9) ^ (1/2)

# y = int1 / sqrt (x ^ 2 + 9) dx #

dať # x = 3 tant ##rArr t = tan ^ -1 (x / 3) #

Z toho dôvodu, # dx = 3sec ^ 2tdt #

# y = int (3sec ^ 2t) / sqrt (9tan ^ 2t + 9) dt #

# y = int (sec ^ 2t) / sqrt (tan ^ 2t + 1) dt #

# y = int (sek ^ 2t) / sqrt (sek ^ 2t) dt #

# y = int (sek ^ 2t) / (sekcia) dt #

# y = int (sekcia) dt #

# y = ln | sek t + tan t | + C #

# y = ln | sec (tan ^ -1 (x / 3)) + tan (tan ^ -1 (x / 3)) | + C #

# y = ln | sec (tan ^ -1 (x / 3)) + x / 3) + C #

# y = ln | sqrt (1 + x ^ 2/9) + x / 3 | + C #

odpoveď:

My to vieme, # int1 / sqrt (X ^ 2 + A ^ 2) dX = ln | X + sqrt (X ^ 2 + A ^ 2) | + c #

takže, # I = int1 / (x ^ 2 + 9) ^ (1/2) dx = int1 / sqrt (x ^ 2 + 3 ^ 2) dx #

# => I = ln | x + sqrt (x ^ 2 + 9) | + c #

vysvetlenie:

# II ^ (nd) # metóda: Trig. subst.

# I = int1 / (x ^ 2 + 9) ^ (1/2) dx #

Zoberme # X = 3tanu => dx = 3 s ^ 2udu #

# a farba (modrá) (tanu = x / 3 #

takže, # I = int1 / (9tan ^ 2u + 9) ^ (1/2) 3sec ^ 2udu #

# = Int (3sek ^ 2U) / ((9 sec ^ 2u) ^ (1/2)) du #

# = Int (3sek ^ 2U) / (3sek) du #

# = Intsecudu #

# = Ln | Secu + nu | + c #

# = Ln | sqrt (tan ^ 2u + 1) + nu | + c #, kde, #COLOR (modro) (nu = x / 3 #

#:. I = ln | sqrt (x ^ 2/9 + 1) + x / 3 | + c #

# = Ln | sqrt (x ^ 2 + 9) / 3 + x / 3 | + c #

# = Ln | (sqrt (x ^ 2 + 9) + x) / 3 | + c #

# = Ln | sqrt (x ^ 2 + 9) + x | -ln3 + c #

# = ln | x + sqrt (x ^ 2 + 9) | + C, kde C = c-ln3 #