Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 decade ago

integrate: e^n/(1+e^(2*n))?

Thank you so much to all who've helped me tonight! I know the answer is tan-1 (e^n)..i'm just unsure how to get there. any help would be greatly appreciated.

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  • Hemant
    Lv 7
    1 decade ago
    Favorite Answer

    Note : INT [ 1 / ( 1 + u^2 ) ] du = tan^-1 ( u )..........(1)

    ...........................................................................................

    [ I am taking x in place of n. ]

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    Let u = e^x.

    Then, du = e^x dx.

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    I = INT [ e^x / ( 1 + e^2x ) ] dx

    = INT { 1 / [ 1 + ( e^x )^2 ] } e^x dx

    = INT [ 1 / ( 1 + u^2 ) ] du,...........where u = e^x

    = tan^-1 ( u ) + C

    = tan^-1 ( e^x ) + C. ..................................Ans.

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  • 1 decade ago

    ∫ e^n / (1 + e^(2n)) dn

    u = e^n

    du = e^n dn

    ∫ 1 / (1 + u²) du

    Remember the rule:

    ∫ 1 / (a² + u²) du = (1/a) arctan (u/a) + C

    So do that for our integral:

    ∫ 1 / (1 + u²) du

    = (1/1) arctan(u/1) + C

    Substitute back in the n's:

    arctan(e^n) + C

    Hope it helps.

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  • 1 decade ago

    make substitution

    y = e^n dy = e^n dn

    substitute to find f(y)dy = dy/(1+y^2) integrate to get tan-1

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