Soalan # f9641

Jawapan:

#int cos (x) / (sin ^ 2 (x) + sin (x)) "d" x = ln | sin (x) / (sin (x)

Penjelasan:

# int cos (x) / (sin ^ 2 (x) + sin (x)) "d" x #

Pengganti # u = sin (x) # dan # "d" u = cos (x) "d" x #. Ini memberi

# = int ("d" u) / (u ^ 2 + u) #

# = int ("d" u) / (u (u + 1)) #

Pisahkan pecahan separa sejak # 1 / (u (u + 1)) = 1 / u-1 / (u + 1) #:

# = int (1 / u-1 / (u + 1)) "d" u #

# = ln | u | -ln | u + 1 | + C #

# = ln | u / (u + 1) | + C #

Gantikan semula # u = sin (x) #:

# = ln | sin (x) / (sin (x) +1) | + C #