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)