1/Cos x'in İntegrali ln |sec x+tan x|'tir.
1/Cos x'in İntegrali Nedir ? ∫ cos x 1 d x = l n ∣ sec x + t an x ∣ + c = − l n ∣ sec x − t an x ∣ + c = l n ∣ 1 − s in x 1 + s in x ∣ + c
1/Cos x'in İntegralini Bulma 1. Yol cos x 1 = sec x ∫ cos x d x = ∫ sec x d x ∫ cos x d x = ∫ sec x + t an x sec x . ( sec x + t an x ) d x
∫ cos x d x = ∫ sec x + t an x se c 2 x + sec x . t an x d x
sec x + t an x = u
d ( sec x + t an x ) = d u
( sec x + t an x ) ′ d x = d u
( sec x ) ′ d x + ( t an x ) ′ d x = d u
( sec x ) ′ = sec x . t an x ( t an x ) ′ = 1 + t a n 2 x = co s 2 x 1 = se c 2 x
sec x . t an x d x + se c 2 x d x = d u
( sec x . t an x + se c 2 x ) d x = d u
∫ cos x d x = ∫ u d u
∫ x d x = l n ∣ x ∣ + c
∫ cos x d x = l n ∣ u ∣ + c
∫ cos x d x = l n ∣ sec x + t an x ∣ + c
∫ cos x d x = l n ∣ sec x − t an x ( sec x − t an x ) ( sec x + t an x ) ∣ + c
∫ cos x d x = l n ∣ sec x − t an x se c 2 x − t a n 2 x ∣ + c
sec x = cos x 1 t an x = cos x s in x ∫ cos x d x = l n ∣ sec x − t an x ( cos x 1 ) 2 − ( cos x s in x ) 2 ∣ + c ∫ cos x d x = l n ∣ sec x − t an x co s 2 x 1 − co s 2 x s i n 2 x ∣ + c ∫ cos x d x = l n ∣ sec x − t an x co s 2 x 1 − s i n 2 x ∣ + c 1 − s i n 2 x = co s 2 x
∫ cos x d x = l n ∣ sec x − t an x co s 2 x co s 2 x ∣ + c
∫ cos x d x = l n ∣ sec x − t an x 1 ∣ + c
∫ cos x d x = l n ∣ ( sec x − t an x ) − 1 ∣ + c
∫ cos x d x = − l n ∣ sec x − t an x ∣ + c
2. Yol ∫ cos x d x = ∫ cos x cos x . cos x 1 d x
∫ cos x d x = ∫ cos x . cos x cos x .1 d x
∫ cos x d x = ∫ co s 2 x cos x d x
co s 2 x = 1 − s i n 2 x
∫ cos x d x = ∫ 1 − s i n 2 x cos x d x
s in x = u
d ( s in x ) = d u
( s in x ) ′ d x = d u
( s in x ) ′ = cos x
cos x d x = d u
∫ cos x d x = ∫ 1 − u 2 d u
∫ cos x d x = ∫ ( 1 − u ) . ( 1 + u ) d u
( 1 − u ) . ( 1 + u ) 1 = 1 − u A + 1 + u B
( 1 − u ) . ( 1 + u ) 1 = ( 1 − u ) . ( 1 + u ) A . ( 1 + u ) + B . ( 1 − u )
( 1 − u ) . ( 1 + u ) 1 = ( 1 − u ) . ( 1 + u ) A + A u + B − B u
0. u + 1 = ( A − B ) . u + ( A + B )
A − B = 0
A + B = 1
A − B + A + B = 0 + 1
2 A = 1
A = 2 1
2 1 + B = 1
B = 1 − 2 1 = 2 1
∫ cos x d x = ∫ 1 − u 1/2 d u + ∫ 1 + u 1/2 d u
∫ cos x d x = 2 1 ∫ 1 − u d u + 2 1 ∫ 1 + u d u
∫ cos x d x = 2 1 ( ∫ 1 − u d u + ∫ 1 + u d u )
∫ a ± x d x = ± l n ∣ a ± x ∣
∫ cos x d x = 2 1 ( − l n ∣1 − u ∣ + l n ∣1 + u ∣ ) + c
∫ cos x d x = 2 1 ( l n ∣1 + u ∣ − l n ∣1 − u ∣ ) + c
∫ cos x d x = 2 1 l n ∣ 1 − u 1 + u ∣ + c
∫ cos x d x = l n ∣ ( 1 − u 1 + u ) 2 1 ∣ + c
∫ cos x d x = l n ∣ 1 − u 1 + u ∣ + c
∫ cos x d x = l n ∣ 1 − s in x 1 + s in x ∣ + c