# 1910fa17-recitation10.pdf - u00a77.8 I NVERSE T RIG...

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§ 7.8: I NVERSE T RIG § 7.9: H YPERBOLIC T RIGONOMETRY § 8.1: I NTEGRATION BY P ARTS N AME : Math 1910 October 17, 2017 O NE - PAGE R EVIEW (1) sinh ( x ) = e x - e - x 2 cosh ( x ) = e x + e - x 2 tanh ( x ) = sinh ( x ) cosh ( x ) coth ( x ) = cosh ( x ) sinh ( x ) sech ( x ) = 1 cosh ( x ) csch ( x ) = 1 sinh ( x ) (2) Derivatives of hyperbolic trigonometric functions d dx sinh ( x ) = cosh ( x ) d dx cosh ( x ) = sinh ( x ) d dx tanh ( x ) = sech 2 ( x ) d dx coth ( x ) = - csch 2 ( x ) d dx sech ( x ) = - sech ( x ) tanh ( x ) d dx csch ( x ) = - csch ( x ) coth ( x ) (3) Integrals of hyperbolic trigonometric functions Z sinh ( x ) dx = cosh ( x ) + C Z cosh ( x ) dx = sinh ( x ) + C Z sech 2 ( x ) dx = tanh ( x ) + C Z csch 2 ( x ) dx = - coth ( x ) + C Z sech ( x ) tanh ( x ) dx = - sech ( x ) + C Z csch ( x ) coth ( x ) dx = - csch ( x ) + C (4) Integration by parts Z u dv = ( 1 ) (5) (Repeat from last Thursday) Derivatives and integrals involving inverse trigonometric functions. d dx sin - 1 ( x ) = 1 1 - x 2 d dx cos - 1 ( x ) = - 1 1 - x 2 d dx tan - 1 ( x ) = 1 x 2 + 1 d dx cot - 1 ( x ) = - 1 x 2 + 1 d dx sec - 1 ( x ) = 1 | x | x 2 + 1 d dx csc - 1 ( x ) = - 1 | x | x 2 + 1 Z 1 1 - x 2 dx = sin - 1 ( x ) + C Z 1 x 2 + 1 dx = tan - 1 ( x ) + C Z 1 | x | x 2 + 1 dx = sec - 1 ( x ) + C 1
P ROBLEMS (1) Simplify sinh ( ln x ) and tanh ( 1 2 ln ( x )) . (2) Find the derivative.