1+sin2x/1+cos2x e²ˣ,Integrate the given function w.r.t. x considering them well defined over proper domains.
Answers
Answered by
4
HELLO DEAR,
GIVEN:-
∫(1 + sin2x)/(1 + cos2x).e²ˣ.dx
put 2x = t
=> 2.dx = dt
therefore,
I = 1/2∫(1 + sint)/{2cos²(t/2}.e^t.dt
=> I = 1/2[∫1/2cos²(t/2) + ∫2sin(t/2)cos(t/2)/2cos²(t/2)]e^t.dt
=> I = 1/2[∫{1/2sec²(t/2) + tan(t/2)}e^t.]dt
we know if integral is in the form of e^x{f(x) + f'(x)}.dx then your integral is e^xf(x) + c.
therefore, I = 1/2e^t{tan(t/2)} + C.
put the value of t = 2x
=> I = 1/2e²ˣtanx + C.
I HOPE ITS HELP YOU DEAR,
THANKS
GIVEN:-
∫(1 + sin2x)/(1 + cos2x).e²ˣ.dx
put 2x = t
=> 2.dx = dt
therefore,
I = 1/2∫(1 + sint)/{2cos²(t/2}.e^t.dt
=> I = 1/2[∫1/2cos²(t/2) + ∫2sin(t/2)cos(t/2)/2cos²(t/2)]e^t.dt
=> I = 1/2[∫{1/2sec²(t/2) + tan(t/2)}e^t.]dt
we know if integral is in the form of e^x{f(x) + f'(x)}.dx then your integral is e^xf(x) + c.
therefore, I = 1/2e^t{tan(t/2)} + C.
put the value of t = 2x
=> I = 1/2e²ˣtanx + C.
I HOPE ITS HELP YOU DEAR,
THANKS
Similar questions