How to Convert an Integral from (1 + x) to 1 - n/(x+n)

[confused88]

Hi! I'm having trouble understanding my textbook, so can someone please explain to me how they got from


$$\int1/(1+x)$$2 dx, with the range of the integral from 0 to x/n

to

1 - n/(x+n)


THank you So MuCh

 

[Pengwuino]

Are you having trouble with the integration or the final answer?

u substitution works fine here, let u = (1 + x) and integrate. The final answer is a simple algebraic trick, nothing spectacular.

 

[confused88]

I'm just having trouble with the integration ><. Oh wells

 

[Pengwuino]

You're looking for the integral of $$\mathop \smallint \nolimits_0^{x/n} \frac{1}{{(1 + x')^2 }}dx'$$ (technically you're not allowed to have your variable of integration in your bounds)

Make a substitution so that your integral now becomes $$\mathop \smallint \nolimits_0^{x/n} \frac{1}{{(u)^2 }}du\frac{{dx}}{{du}}$$ and remember to compute dx/du.

What, when you take its derivative becomes $$\frac{1}{{u^2 }}$$? Find what that is, substitute back in for what you had set u to and you can plug in your integration limits and wala!