Evaluate the improper integral using a suitable

Question

1. (a) Given that sinh(x), cosh(x) and tanh(x) are defined as follows

cosh(x) = 1/2 (e^x + e^-x), sinh(x) = 1/2(e^x - e^-x),   tanh(x) = sinh(x)/cosh(x)

show that

tanh^-1(x) = 1/2 log_e (1+x/1-x)

(b) Compute the following derivative

d/dx(cosh^-1(tanh(2x)))

(c) Evaluate the following improper integral using a suitable substitution. Be sure to treat the improper integral with due care.

I = _-∞∫^3Π 4/4+x²dx

2. (a) Use the definitions

cosh(x) = 1/2(e^x + e^-x), sinh(x) = 1/2(e^x - e^-x)

to express sinh(x + y) and cosh(x + y) in terms of cosh(x), sinh(x), cosh(y) and sinh(y).

(b) Using the results of part (a) show that

sinh(mx + x) = cosh(mx) sinh(x) + sinh(mx) cosh(x)
cosh(mx + x) = cosh(mx) cosh(x) + sinh(mx) sinh(x)

(c) Use the result of part (b) to express the following sums

C_n = cosh(1x) + cosh(2x) + cosh(3x) +


cosh(nx)
S_n = sinh(1x) + sinh(2x) + sinh(3x) +


sinh(nx)

in terms of just cosh((n + 1)x), sinh((n + 1)x), cosh(x) and sinh(x) (and possibly some numbers like 1, 2, e etc.).