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1E1 Hilary Term exam 2005 1. Calculate the following integrals, using substitution or integration by parts as appropriate. Z 2x √ dx (a) x2 + 1 Z (b) e2x cos x dx Z 3 (c) x2 e(x ) dx Z e ln x dx (d) Z1 tan x (e) dx cos x 2. Calculate the following integrals and derivatives. Z 1 (a) dx 2(x + 4) ¡ ¢ d (b) tanh x4 Zdx (c) (d) (e) x sinh x dx d ln (cosh x) dx Z cosh x dx sinh2 x 3. Solids of revolution. (a) Consider the functions f and g defined by f (x) = x and g(x) = x2 . Calculate the volume of the solid of revolution generated by revolving the area bounded by the graphs of f and g about the x-axis. (b) Consider the function h defined by h(x) = x − 2. Calculate the volume of the solid of revolution generated by revolving the area bounded by the x-axis, the graph of h and the line x = 3 about the line x = 1. 1 4. Derivatives of inverse functions. (a) Show that d 1 coth−1 x = . dx 1 − x2 (b) Let f be a function that is defined on an open interval D. Let f be differentiable with f 0 (x) 6= 0 for all x in D. Then f is one-to-one and thus invertible. Let R be the range of f . Show that (f −1 )0 (y) = with y = f (x), for all y in R. 2 1 f 0 (x)