Tan x'in integrali -ln |cos x|'tir.
Tan x'in İntegrali Nedir ? Tan x'in integrali -ln |cos x|'tir.
∫ t an x d x = − l n ∣ cos x ∣ + c = l n ∣ sec x ∣ + c
Tan x'in İntegralini Bulma 1. Yol ∫ t an x d x = ? t an x = cos x s in x
∫ t an x d x = ∫ cos x s in x d x
cos x = u
d ( cos x ) = d u
( cos x ) ′ d x = d u
( cos x ) ′ = − s in x
− s in x d x = d u
s in x d x = − d u
∫ t an x d x = ∫ u − d u
∫ t an x d x = − ∫ u d u
∫ x d x = l n ∣ x ∣ + c
∫ t an x d x = − l n ∣ u ∣ + c
∫ t an x d x = − l n ∣ cos x ∣ + c
∫ t an x d x = l n ∣ co s − 1 x ∣ + c
∫ t an x d x = l n ∣ cos x 1 ∣ + c
cos x 1 = sec x
∫ t an x d x = l n ∣ sec x ∣ + c
2. Yol ∫ t an x d x = ∫ sec x sec x . t an x d x
∫ t an x d x = ∫ sec x sec x . t an x d x sec x = u d ( sec x ) = d u ( sec x ) ′ d x = d u ( sec x ) ′ = sec x . t an x sec x . t an x d x = d u
∫ t an x d x = ∫ u d u
∫ t an x d x = l n ∣ u ∣ + c
∫ t an x d x = l n ∣ sec x ∣ + c
3. Yol
Yukarıdaki ABC dik üçgeninde;
t an x = 1 u = u
∫ t an x d x = ?
t an x = u
d ( t an x ) = d u
( t an x ) ′ d x = d u
( t an x ) ′ = 1 + t a n 2 x
( 1 + t a n 2 x ) d x = d u
( 1 + u 2 ) d x = d u
d x = 1 + u 2 d u
∫ t an x d x = ∫ u . 1 + u 2 d u
∫ t an x d x = ∫ 1 + u 2 u d u
∫ t an x d x = ∫ 2. ( 1 + u 2 ) 2. u d u
∫ t an x d x = 2 1 ∫ 1 + u 2 2 u d u
1 + u 2 = v
d ( 1 + u 2 ) = d v
( 1 + u 2 ) ′ d u = d v
2 u d u = d v
∫ t an x d x = 2 1 ∫ v d v
∫ t an x d x = 2 1 l n ∣ v ∣ + c
∫ t an x d x = l n ∣ v 2 1 ∣ + c
∫ t an x d x = l n ∣ v ∣ + c
∫ t an x d x = l n ∣ 1 + u 2 ∣ + c
Yukarıdaki ABC dik üçgeninde;
sec x = 1 1 + u 2 = 1 + u 2
∫ t an x d x = l n ∣ sec x ∣ + c