Anonymous

# 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.

Relevance
• Hemant
Lv 7

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

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[ 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.

∫ 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.