perplexus.info :: Calculus : Exponential And Natural

Exponential And Natural (Posted on 2008-01-08)

Determine the value of a real constant c, given that:

     _y
   ∫(ln p)*(1+p)^-1 dp = g(y)
    ¹

and, g(e^c) + g(e^-c) = c³/12

where ln x denotes the natural logarithm of x.

Submitted by K Sengupta

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Solution: (Hide)

The required value of c is 0 or 6.

EXPLANATION:

integral (1+p)^-1* ln p dp, p = 1 to y is denoted by I₁.

Now, consider the integral:

I₂ = integral (1+p)^-1* ln p dp, p = 1 to 1/y

Substituting p =1/q and simplifying, we have:

I₂ = integral (ln q)/q dq, q = 1 to y - integral (1+q)^-1(ln q) dq, q = 1 to y

= (ln y)²/2 - I₁.

Thus, g(y) + g(1/y) = (ln y)²/2 ЕЕЕ..(*)

Since c is a real constant, in terms of (*) we must have:

g(e^c) + g(e^-c) = c²/2

Accordingly, from the second given relationship, we obtain:

c²/2 = c³/12, so that:

c = 0 or 6

.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
question!	Chesca Ciprian	2008-01-10 16:31:36
full solution	Daniel	2008-01-09 09:44:22
not sure!	Chesca Ciprian	2008-01-09 06:21:29

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