Q ∫tan xa+b tan2 xdx - Maths - Integrals

Question

Q   ∫ tan   x a + b   tan 2   x d x

Shruti Tyagi · Accepted Answer

Dear student,

Let I=∫tanxa+btan2xdx=∫sinxcosxa+bsin2xcos2xdx=∫sinxcosxacos2x+bsin2xcos2xdx             =∫sinxcosxacos2x+bsin2xdx=∫sinxcosxa1-sin2x+bsin2xdx             --[using cos2x=1-sin2x in Dn]=∫sinxcosxa-asin2x+bsin2xdxI=∫sinxcosxa+b-asin2xdx         --(1)Put a+b-asin2x=u⇒2b-asinxcosx dx=du⇒sinxcosx dx=du2b-afrom (1)I=12(b-a)∫1uduI=12(b-a)log (u)+CUndo substitution u=a+b-asin2xI=12(b-a)log a+b-asin2x+C

Regards

Q . ∫ tan x a + b tan 2 x d x

$Q . \int \frac{\tan x}{a + b \tan^{2} x} d x$