Question

ayuda!! Obtener integral de las siguientes funciones trigonométricas
∫sen^2 (x) dx
∫cos^2⁡ (3x) dx
∫sen^3⁡ (x) dx
∫sec^4⁡ (2x) dx

Answer 1

a) integral de sen^2(x) dx = integral de (2sen^2(x) /2)dx
integral de ((1 - cos 2x)/2)dx = integral de( 1/2 - cos2x/2)dx
= x/2 - 1/2 integral de cos2xdx + c
x/2 -1/2 (sen2x/2) + c
x/2 - sen2x/4 + c

b) integral de cos^2(3x)dx = integral de ((2cos^2(3x))/2 dx
integral de ((1 + cos6x)/2)dx = integral de (1/2 + cos6x/2)
x/2 + integral de cos6x/2 + c = x/2 + 1/2 (integral de cos6x)
= x/2 + 1/2 (-sen6x/6) + c
= x/2 - sen6x/12 + c

c) integral de sen^3 (x)dx = integral de sen^2(x) senxdx
= integral de (1 - cos^2(x))senxdx
= integral de [(senx) - cos^2(x)senx]dx
= integral de senxdx - integral de cos^2(x)senxdx
-cosx - (-cos^3(x)/3) + c
-cosx + (cos^3(x))/3 + c

d) integral de sec^4 (2x)dx = integral de sec^2(x) sec^2(x)
= integral de (1+ tg^2(x)) sec^2(x) dx
= integral de [sec^2(x) + tg^2(x) sec^2(x)] dx
= integral de sec^2(x)dx + integral de tg^2(x) sec^2(x) dx
= tg(x) + integral de tg^2(x) sec^2(x) dx +c
Sea u = tg(x) ======》du = sec^2(x)dx
Entonces
=tg(x) + integral de (u^2) du + c
= tg(x) + (u^3)/3 + c
= tg(x) + (tg^3(x))/3 + c