¿Demostrar la formula de reduccion de ∫tan^n x*sec^m x dx = ( tan^(n-1) x sec^(m-2) x ) / (n+m-1) + (m-2) / (n+m-1) ∫tan^nx sec^(m-2) x dx?
Update:
por favor si alguien sabe como demostrar esto le agradecería mucho me ayudara con esto ......
More Questions From This User See All
Answers & Comments
Verified answer
Hola
sec^2(x) = 1 + tan^2(x)
d(tan(x)) = sec^2(x) dx
d(sec(x)) = tan(x) sec(x) dx
∫ tan^(n)(x) sec^(m)(x) dx = ∫ tan^n(x) sec^(m-2)(x) (sec^2(x) dx)
u = tan^(n)(x) sec^(m-2)(x)
dv = sec^2(x)
-> v = tan(x)
du = (n) tan^(n-1)(x) sec^(m-2)(x) d(tan(x)) +
+ (m-2) tan^(n)(x) sec^(m-3)(x) d(sec(x))
du = (n) tan^(n-1)(x) sec^(m-2)(x) sec^(2)(x) dx +
+ (m-2) tan^(n)(x) sec^(m-3)(x) tan(x) sec(x) dx
du = (n) tan^(n-1)(x) sec^(m)(x) dx +
+ (m-2) tan^(n+1)(x) sec^(m-2)(x) dx
∫ tan^(n)(x) sec^(m)(x) dx = u v - ∫v du
∫ tan^(n)(x) sec^(m)(x) dx = (tan^n(x) sec^(m-2)(x)) (tan(x)) -
- ∫tan(x) (n) tan^(n-1)(x) sec^(m)(x) dx -
- ∫tan(x) (m-2) tan^(n+1)(x) sec^(m-2)(x) dx
∫ tan^(n)(x) sec^(m)(x) dx = (tan^(n+1)(x) sec^(m-2)(x)) -
- (n) ∫ tan^(n)(x) sec^(m)(x) dx -
- (m-2) ∫ tan^(n+2)(x) sec^(m-2)(x) dx
∫ tan^(n)(x) sec^(m)(x) dx = (tan^(n+1)(x) sec^(m-2)(x)) -
- (n) ∫ tan^(n)(x) sec^(m)(x) dx -
- (m-2) ∫ tan^(n)(x) sec^(m-2)(x) (tan^2(x))dx
∫ tan^(n)(x) sec^(m)(x) dx = (tan^(n+1)(x) sec^(m-2)(x)) -
- (n) ∫ tan^(n)(x) sec^(m)(x) dx -
- (m-2) ∫ tan^(n)(x) sec^(m-2)(x) (sec^2(x) - 1)dx
∫ tan^(n)(x) sec^(m)(x) dx =
= (tan^(n+1)(x) sec^(m-2)(x)) -
- (n) ∫ tan^(n)(x) sec^(m)(x) dx -
- (m-2) ∫ tan^(n)(x) sec^(m)(x) dx +
+ (m-2) ∫ tan^(n)(x) sec^(m-2)(x) dx
*****************************************************
Despejamos la integral original
(1 + (n) + (m - 2)) ∫ tan^(n)(x) sec^(m)(x) dx =
= (tan^(n+1)(x) sec^(m-2)(x)) +
+ (m-2) ∫ tan^(n)(x) sec^(m-2)(x) dx
(n + m - 1) ∫ tan^(n)(x) sec^(m)(x) dx =
= (tan^(n+1)(x) sec^(m-2)(x)) +
+ (m-2) ∫ tan^(n)(x) sec^(m-2)(x) dx
∫ tan^(n)(x) sec^(m)(x) dx =
= (1/(n + m - 1)) tan^(n+1)(x) sec^(m-2)(x) +
+ ((m - 2)/(n + m - 1)) ∫ tan^(n)(x) sec^(m-2)(x) dx
*******************************************************
Saludos
ufff... maestro jejej muchísimas gracias ^-^ !!!!