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Math Help - Definite integral

  1. #1
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    Definite integral

    Show that:
    the integral between 1 and 0 of(1+(a-1)x)^n= (a^(n+1)-1)/{(n+1)(a-1)}

    where a isn't 1 and n is a positive integer.

    Please help. I have no idea how to intergrate it.

    Thanks
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by free_to_fly View Post
    Show that:
    the integral between 1 and 0 of(1+(a-1)x)^n= (a^(n+1)-1)/{(n+1)(a-1)}

    where a isn't 1 and n is a positive integer.

    Please help. I have no idea how to intergrate it.

    Thanks
    First, let's do the integral: (note, you just treat a - 1 as a constant and do it as you would any other integral)

    int{(1+(a-1)x)^n}dx

    We proceed by substitution:
    Let u = 1+(a-1)x
    => du = (a - 1) dx
    => 1/(a - 1)du = dx
    so our integral becomes:
    1/(a - 1)*int{u^n}du
    = 1/(a - 1){u^(n + 1)]/(n + 1)} + C
    = 1/(a - 1){[(1 + (a - 1)x)^(n + 1)]/(n + 1)} + C
    Now we evaluate our answer between 0 and 1.
    1/(a - 1)*{[(1 + (a - 1))^(n + 1)/(n + 1)] - [(1 + (a - 1)(0))^(n + 1)/(n + 1)]}
    = 1/(a -1)*{a^(n+1)/(n+1) - 1^(n+1)/(n+1)}
    = 1/(a -1)*{a^(n+1)/(n+1) - 1/(n+1)}
    = 1/(a - 1)*((a^(n+1) - 1)/(n + 1))
    = (a^(n+1)-1)/{(n+1)(a-1)} as desired
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  3. #3
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    Thanks

    Thanks so much for your help. Now I've avoided the shrill screams from my maths teachers. Thanks
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