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$\int \frac{(x^2 + x)}{(e^x + x + 1)^2}dx$

I was thinking along the lines of breaking the numerator into denominator differentiation and generic function with help of division rule.

K.K.McDonald
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Mrigank
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1 Answers1

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HINT:

Divide the numerator & denominator by $e^{2x}$

$$\dfrac{x^2+x}{(e^x+x+1)^2}=\dfrac{xe^{-x}\cdot(x+1)e^{-x}}{\{1+e^{-x}(x+1)\}^2}$$

Set $1+e^{-x}(x+1)=u$

lab bhattacharjee
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  • ok, granted, is there a better {more general} way to reach it, suppose coefficients were not as simple as all 1's, is there a general way to approach these type of questions where e^x and polynomials are mixed? – Mrigank Mar 27 '16 at 16:23
  • @ELiT generally no, for example, there is no closed form for the integral $\int \frac{e^x}{x}dx$ – lEm Mar 27 '16 at 16:24
  • I'm talking about exactly the same type of questions with just different coefficients , given that it is solvable and also quite easy with one or two step problem like the problem given above, How should we go about breaking the numerator {and then finding values like we do in partial fractions} or is there a better method? – Mrigank Mar 27 '16 at 16:34