AP EAMCET Engineering 22 Sep 2020 Shift 2 Paper

∫‌

5‌tan‌x

tan‌x−2

dx=∫‌

5‌sin‌x

sin‌x−2‌cos‌x

dx
Put, On comparison of coefficients of sin‌x,cos‌x we get
sin‌x coefficients →5=α+2β...(i)
cos‌x coefficients →0=−2α+β
On solving Eqs. (i) and (ii), weget
α=1 and β=2
Now, substitute 5‌sin‌x value in the given question.
∫‌

5‌sin‌x

(sin‌x−2‌cos‌x)

dx
|∵∫‌

f′(x)

f(x)

dx=log|f(x)||
=x+2‌log|sin‌x−2‌cos‌x|+c
On comparison with given RHS, we get
α=1,β=2⇒α−β=−1