Question

please give answer and solution of this​

Answer 1

Answer:

1/10 log ( 4 Cos²x + 9 Sin²x ) + C

Step-by-step explanation:

To find ----->

∫ tanx / ( 4 + 9tan²x ) dx = ................... + C

Solution------> 1) Plz refer the attachement

2) First , we use a formula

tanθ = Sinθ / Cosθ

then , we take LCM , in denominator and we get

I = ∫ Sinx Cosx dx / ( 4 Cos²x + 9 Sin²x )

3) Then we suppose ,

4 Cos²x + 9 Sin²x = t

Differentiating with respect to x both sides , by using three formulee of differentiation ,

d / dx ( x² ) = 2x

d / dx ( Sinx ) = Cosx

d / dx ( Cosx ) = - Sinx

{ 8 Cosx ( - Sinx ) + 18 Sinx Cosx } dx = dt

10 Sinx Cosx dx = dt

Sinx Cosx dx = dt / 10

and we get ,

I = 1/10 ∫ dt/t

Now , we have a formula of integration ,

∫ dx / x = log x + C

= 1/10 ( log t ) + C

= 1/10 log ( 4Cos²x + 9Sin²x ) + C