invhyp Derivatives, Integrals, and Properties Of Inverse Trigonometric Functions and Hyperbolic Functions

Derivatives, Integrals, and Properties Of Inverse Trigonometric Functions and Hyperbolic Functions (On this handout, a represents a constant, u and x represent variable quantities) Derivatives of Inverse Trigonometric Functions Identities for Hyperbolic Functions d sin¡1 u dx sinh 2x = sinh x cosh x d cos¡1 u dx = p = p du ¡ u dx ¡1 du ¡ u2 dx d tan¡1 u = dx du + u2 dx d csc¡1 u dx = d sec¡1 u dx ¡1 du p juj u2 ¡ dx = d cot¡1 u dx = du p juj u2 ¡ dx (juj < 1) cosh 2x = cosh2 x + sinh2 x (juj < 1) Z Z p a2 a2 du ¡ u2 (juj > 1) ¡1 du + u2 dx du + u2 = sin¡1 = p du = u u ¡ a2 ³u´ +C a ³u´ tan¡1 +C a a ¯u¯ ¯ ¯ sec¡1 ¯ ¯ + C a a sinh x = ¡x x ¡x e ¡e sinh2 x = cosh 2x ¡ cosh x = e +e x = sinh x ex ¡ e¡x = x cosh x e + e¡x cschx = = x sinh x e ¡ e¡x sechx = = x cosh x e + e¡x coth x = cosh x ex + e¡x = x sinh x e ¡ e¡x cosh2 x ¡ sinh2 x = tanh2 x = ¡ sech2 x coth2 x = + csch2 x Derivatives of Hyperbolic Functions (Valid for u2 < a2 ) d sinh u dx = cosh u (Valid for all u) d cosh u dx = sinh u (Valid for u2 > a2 ) d u = dx The Six Basic Hyperbolic Functions x cosh 2x + (juj > 1) Integrals Involving Inverse Trigonometric Functions Z cosh2 x = du dx du dx sech2 u du dx du dx d coth u dx = ¡ csch2 u d sechu dx = ¡ sechu u d cschu dx = ¡ cschu coth u du dx du dx Inverse Hyperbolic Identities µ ¶ sech x = cosh x µ ¶ ¡1 ¡1 csch x = sinh x µ ¶ coth¡1 x = tanh¡1 x ¡1 ¡1 Integrals Involving Inverse Hyperbolic Functions Integrals of Hyperbolic Functions Z Z Z Z Z Z sinh u du Z = cosh u + C cosh u du Z = sinh u + C sech u du = u + C csch2 u du = ¡ coth u + C du a2 + u2 = sinh¡1 p du u2 ¡ a2 = cosh¡1 = ¡ cschu + C d cosh¡1 u dx = = d tanh¡1 u = dx d csch¡1 u dx = d sech¡1 u dx = d coth¡1 u dx = p du + u2 dx du p u ¡ dx (u > 1) du ¡ u2 dx (juj < 1) ¡1 du p juj + u dx (u 6= 0) ¡1 du p u ¡ u dx du ¡ u2 dx p du u § a2 p x2 + 1) cosh¡1 x = ln(x + p x2 ¡ 1) tanh¡1 x = du a ¡ u2 +C (a > 0) +C (u > a > 0) sech¡1 x csch¡1 x coth¡1 x = ln(u + p (¡1 < x < 1) (x ¸ 1) 1+x ln (jxj < 1) 1¡x à ! p + ¡ x2 = ln (0 < x · 1) x à ! p 1 + x2 = ln + (x 6= 0) x jxj = x+1 ln x¡1 u2 § a2 ) + C ¯ ¯ ¯a + u¯ ¯ ¯+C = ln 2a ¯ a ¡ u ¯ à ! p Z 1 a + a2 § u2 p du = ¡ ln +C a juj u a2 § u2 Z a = ln(x + Alternate Form For Integrals Involving Inverse Hyperbolic Functions Z ³u´ sinh¡1 x (0 < u < 1) (juj > 1) a Expressing Inverse Hyperbolic Functions As Natural Logarithms Derivatives of Inverse Hyperbolic Functions d sinh¡1 u dx ³u´ ³ ´ ¡1 u > > + C (if u2 < a2 ) > > Z a a < ³ ´ du = ¡1 u > a ¡ u2 > coth + C (if u2 > a2 ) > > a : a Z ³ ´ 1 ¡1 u p du = ¡ sech + C (0 < u < a) a a u a2 ¡ u2 Z ¯u¯ 1 ¯ ¯ p du = ¡ csch¡1 ¯ ¯ + C 2 a a u a +u sechu u du = ¡ sechu + C cschu coth u du p (jxj > 1) ... Integrals Involving Inverse Hyperbolic Functions Z ³u´ sinh¡1 x (0 < u < 1) (juj > 1) a Expressing Inverse Hyperbolic Functions As Natural Logarithms Derivatives of Inverse Hyperbolic Functions d sinh¡1...Integrals Involving Inverse Hyperbolic Functions Integrals of Hyperbolic Functions Z Z Z Z Z Z sinh u du Z = cosh u + C cosh u du Z = sinh u +