I am having trouble figuring out an algebraic trick to make this work

Evaluate the integral


I know I can turn the integrand into $(x-1) (1/\sqrt{x})$ but I still don't know how to do products of intregrals.

  • 4
    $\begingroup$ break the integrand into $\sqrt{x}-\frac{1}{\sqrt{x}}$ or substitute $x=u^2$. $\endgroup$ – robjohn Nov 3 '11 at 21:44

Hint: break the integral into $\sqrt{x}-\frac{1}{\sqrt{x}}$ or try substituting $x=u^2$.

  • $\begingroup$ I got $x^.5 - x^-/5$ and the answer is wrong. $\endgroup$ – toby yeats Nov 3 '11 at 22:40
  • $\begingroup$ You still have to integrate $\sqrt{x}-\frac{1}{\sqrt{x}}$. $\endgroup$ – robjohn Nov 3 '11 at 22:48
  • $\begingroup$ Ah you mean find what function has that as a derivative? I forgot to. $\endgroup$ – toby yeats Nov 3 '11 at 23:58

You cannot make the integrand $x-1(1/\sqrt{x})$; you can make it $(x-1)(1/\sqrt{x})$. Those parentheses are important. However, you don’t want to do any such thing. You want to divide it out. Rewrite the integrand as $$\frac{x-1}{\sqrt{x}} = \frac{x}{\sqrt{x}}-\frac1{\sqrt x}$$ and simplify each term to a power of $x$. Then integrate term by term.


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