18 Exercises 4. Final test 1. i) Differentiate with respect to x. y sinh ly + 3x =4 ay -3 (Answer: dx = 1+7 2 =+ sinh- ii) Evaluate dx V1 + 9x2 (Answer: = sinh-13x + c) 2. i) Differentiate with respect to X. sin x y = In sinh-1(2x + 1) [hint: In () = In a - In b] (Answer: cos x - 2 sinh-1 (2x+1)/1+(2x+1)2) dx ii) J V1-e 2x (Answer: -sech-1(ex) + c) m/s 12mis 12 diff 3. i) Differentiate implicitly tanh(x +y) = e2x sinh-ly3 (Answer:- 2e2x sinh ly3-sech?(xty) sech2(xty)-3ezxy2 1+16 ii) Using the completing square method, evaluate dx Vx2 + 2x +3 (Answer: 0.8381)3.4. Integration of the inverse hyperbolic functions Formula: du a) J Vaz+ u2 = sinh-1 (") + cor In (u + Vuz + a2 ) + c du b ) J Vuz-az = cosh 1 (") + c or In(u + Vuz + a2 ) + c tanh-1 du ") + c, lul a du d) J = uva 2 - 42 - sech-1 () + cor = In a+va2-u2 + c du a+va2+uz = csch-1 + cor -In + c e) J uva2+u2 a