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I'm having trouble understanding some of the notation for definite integrals. Is this right?

$$ \int_1^2 (\frac{1}{x^2} - \frac{4}{x^3}) \, dx$$

$$ \int_1^2 x^{-2} - 4x^{-3} \, dx$$

$$ = \int_1^2 x^{-2} \, dx- \int_1^2 4x^{-3} \, dx$$

$$ \left[ -x^{-1} \right ]_1^2 - \left[\frac{4x^{-2}}{-2} \right]_1^2$$

Is this notation right? Or should it be:

$$\left[ -x^{-1} -\frac{4x^{-2}}{-2} \right]_1^2 $$

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  • $\begingroup$ It looks good either way. Note that $$[f(x)-g(x)]_a^b = (f(b)-g(b))-(f(a)-g(a)) = (f(b)-f(a)) - (g(b)-g(a)) = [f(x)]_a^b - [g(x)]_a^b$$ $\endgroup$ – Michael Apr 13 '18 at 5:35
  • $\begingroup$ Second line should be $=\int_1^2(x^{-2}-4x^{-3})\,dx$. $\endgroup$ – Lord Shark the Unknown Apr 13 '18 at 5:38
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It's fine. Lord Shark is correct that the second line is better written

$$\int_1^2 \left(x^{-2} - 4x^{-3}\right) \, dx$$

since the length form $dx$ multiplies the entire function $x^{-2} - 4x^{-3}$ and not just the right term, but the potential for confusion in this case is nil.

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  • $\begingroup$ What about my last 2 lines? $\endgroup$ – Jwan622 Apr 13 '18 at 5:47
  • $\begingroup$ They're fine too. $\endgroup$ – user7530 Apr 13 '18 at 5:47
  • $\begingroup$ Both are acceptable notation? $\endgroup$ – Jwan622 Apr 13 '18 at 5:50
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    $\begingroup$ I don't know how many different ways I can say "yes it's fine" ;) $\endgroup$ – user7530 Apr 13 '18 at 5:57

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