∫ (tanx) / (a+b tan²x) dx , plz answer asap, thank you.

First write down the whole fuction using sines and cosines, then you will notice that the you can convert this as

the derivarive of denominator is equal to the numerator. 

 

 

a  = a/b

b

 

 

int           sin(x)/cos(x)

         a + b.sin^2(x)/cos^2(x)

 

int             sin(x).cos(x)

          a.cos^2(x) + b.sin^2(x)

 

Let, a.cos^2(x) + b.sin^2(x) = t    ….........................eq ( i ) 

 

      a.cos(x)[-sin(x)].dx + b.2sin(x).cos(x).dx = dt 

 

      cos(x).sin(x).dx = dt.½.(b – a) ….......................eq ( ii )

 

 

 

 Therefore substituting the value of eq (i) and eq (ii) in the above question

 

int     [1/t.   dt.1/2(b – a)]

 

1/2(b – a)   int   dt/t      ….......................Since 1/2(b – a) is a constant

 

1/2(b – a) .  [loge|t|  + c]

 

1/2(b – a).loge |a.cos^2(x) + b.sin^2(x)| + c  …............ is the answer