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