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    Hey!

    Was wondering if anyone could help me with this core 3 question:

    Show that when you integrate [ e^x - e^(-x) ] between the limits of one and zero you get:

    [(e - 1)^2]/e

    And these two core 4 questions:

    1. Integration of (e^x)/(1+e^2x) using x=lnu as a substitution

    2. Integration of 1/ (1-x^2)^3/2 using substitution x=sinu

    Thankyou
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    ∫e^x - e^(-x) dx

    = [ex + e-x ] between 0,1

    = e + 1/e - (1 + 1)

    = (e² + 1 -2e)/e

    = (e -1)²/e

    1. Integration of (e^x)/(1+e^2x) using x=lnu as a substitution

    ∫(e^x)/(1+e^2x) dx

    x = lnu

    dx/du = 1/u

    ex = u

    => ∫ u/(1 +u²)(1/u) du

    = ∫ 1/(1 + u²) du.....I'll leave the rest, because it will get tedious and I assume you're allowed to quote this result anyway.

    2. Integration of 1/ (1-x^2)^3/2 using substitution x=sinu

    ∫1/ (1-x^2)^3/2 dx

    x=sinu

    dx/du = cosu = √(1 - sin²u)

    = (1 - x²)1/2

    => (1 - x²)1/2/(1 - x²)3/2

    = (1 - x²)-1 => ∫sec²u du = tanu ... but what's this in terms of x?
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    (Original post by Looontag)
    Hey!

    Was wondering if anyone could help me with this core 3 question:

    Show that when you integrate [ e^x - e^(-x) ] between the limits of one and zero you get:

    [(e - 1)^2]/e

    And these two core 4 questions:

    1. Integration of (e^x)/(1+e^2x) using x=lnu as a substitution

    2. Integration of 1/ (1-x^2)^3/2 using substitution x=sinu

    Thankyou
    ∫ e^x - e^(-x) dx = [e^x +1/e^x] between 1 and 0
    [e + 1/e] - [1+1]
    e+1/e - 2 = e²/e + 1/e -2e/e = (e²-2e+1)/e = (e-1)²/e

    i dont have the time to do the other ones because im revising economics :mad:
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    dammit.. posted too late
 
 
 
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