inverse and hyperbolic

inverse and hyperbolic - Inverse Trigonometric...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Inverse Trigonometric Function/Hyperbolic Function Identities for Inverse Trigonometric Function: Identities for Hyperbolic Function: cos x + cos (− x) = π −1 −1 sinh 2 x = 2sinh x cosh x sin −1 x + cos −1 x = π / 2 sec −1 x = cos −1 (1/ x) csc −1 x = sin −1 (1/ x) cot −1 x = π / 2 − tan −1 x 1 − cos 2 x sin 2 x = 2 1 + cos 2 x cos 2 x = 2 Derivatives of the Inverse Trigonometric Function: cosh 2 x = cosh 2 x + sinh 2 x cosh 2 x + 1 cosh 2 x = 2 cosh 2 x − 1 sinh 2 x = 2 2 2 cosh x − sinh x = 1 tanh 2 x = 1 − sec h 2 x coth 2 x = 1 + csc h 2 x Derivatives of the Inverse Hyperbolic Function: d (sin u ) du / dx = , u <1 dx 1− u2 d (cos −1 u ) du / dx =− , u <1 dx 1− u2 d (tan −1 u ) du / dx = dx 1+ u2 d (cot −1 u ) du / dx =− dx 1+ u2 d (sec −1 u ) du / dx = , u >1 dx u u2 −1 d (csc −1 u ) du / dx =− , u >1 dx u u2 −1 Integration Formulas for Inverse Trig Functions: −1 d (sinh −1 u ) du / dx = dx 1+ u2 d (cosh −1 u ) du / dx = ,u > 1 dx u2 −1 d (tanh −1 u ) du / dx = , u <1 dx 1− u2 d (coth −1 u ) du / dx = , u >1 dx 1− u2 d (sec h −1u ) du / dx =− , 0 < u <1 dx u 1− u2 d (csc h −1u ) du / dx =− ,u ≠ 0 dx u 1+ u2 Integration Formulas for Inverse Hyperbolic Functions: u = sin −1 + C , (valid for u 2 < a 2 ) a a −u du 1 −1 u ∫ a 2 + u 2 = a tan a + C , (valid for all u ) ∫ du 2 2 ∫ ∫ 1 −1 u 2 2 ∫ u u 2 − a 2 = a sec a + C , (valid for u > a ) du u = sinh −1 + C , a > 0 a a +u du u = cosh −1 + C , u > a > 0 2 2 a u −a du 2 2 1 −1 u 2 2 a tanh a + C , u < a du ∫ a 2 − u 2 = 1 −1 u 2 2 coth + C, u > a a a Identities for Inverse Hyperbolic Function: sec h −1 x = cosh −1 1 x 1 csc h −1 x = sinh −1 x 1 coth −1 x = tanh −1 x d (sinh u ) du = cosh u dx dx d (cosh u ) du = sinh u dx dx d (tanh u ) du = sec h 2u dx dx d (coth u ) du = − csc h 2u dx dx d (sec hu ) du = − sec hu tanh u dx dx d (csc hu ) du = − csc hu coth u dx dx Integration Formulas for Inverse Hyperbolic Functions: 1 u = − sech −1 + C , 0 < u < a a a a2 − u 2 du 1 −1 u ∫ u a 2 + u 2 = − a csch a + C , u ≠ 0 ∫u du Derivatives of Hyperbolic Function: Integral Formulas for Hyperbolic Function: ∫ sinh udu = cosh u + C ∫ cosh udu = sinh u + C ∫ sec h udu = tanh u + C ∫ csc h udu = − coth u + C ∫ sec hu tanh u = − sec hu + C ∫ csc hu coth udu = − csc hu + C 2 2 Hyperbolic cosine of x: Hyperbolic secant: cosh x = Hyperbolic sine of x: e +e 2 x −x sec hx = 1 2 = x −x cosh x e + e sinh x = Hyperbolic tangent: e x − e− x 2 Hyperbolic cosecant: csc hx = 1 2 = x −x sinh x e − e cosh x e x + e − x = sinh x e x − e − x sinh x e x − e − x tanh x = = cosh x e x + e − x 2 2 Hyperbolic cotangent: coth x = Trig Substitution a + x ⇒ x = a tan θ ⇒ a 2 + x 2 = a 2 sec 2 θ a 2 − x 2 ⇒ x = a sin θ ⇒ a 2 − x 2 = a 2 cos 2 θ x 2 − a 2 ⇒ x = a secθ ⇒ x 2 − a 2 = a 2 tan 2 θ ...
View Full Document

This note was uploaded on 03/22/2009 for the course MATH M112 taught by Professor Levy during the Fall '05 term at NJIT.

Ask a homework question - tutors are online