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