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(a) Find the solution of the initial-value problem and determine its interval of existence (i.e

domain of the resulting function y).

(i) dy

dt =

1+t

2

t

, y(t = 1) = 0.

(ii) (t + 1) dy

dt = 1 − y, y(t = 0) = 3. [Hint: Let A = −(±e

−C).]

(b) Evaluate the following integrals:

(i) Z

x ln(x + 1)dx

(ii) Z

sin3

(ln x) cos2

(ln x)

x

dx.