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I've learned both methods but I'm struggling with when to know when to sue each one.

Example, current problem: $(x^3 + x) \sqrt{x}$. U substitution isn't working, and when I try parts method I get a messy second integral.

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  • $\begingroup$ What is U substitution? I have never heard of it. (I suspect it's a newfangled name for something I do know.) $\endgroup$ – Harald Hanche-Olsen Oct 5 '14 at 22:28
  • $\begingroup$ @HaraldHanche-Olsen You may also know it as the anti-chain rule, but most people call it the substitution method or u-substitution. $\endgroup$ – user137731 Oct 5 '14 at 22:31
  • $\begingroup$ @user3032755 I wouldn't use either in this case (see mathlove and Paul's hints), but in general substitution is a good method when a half of the integrand is very similar to the derivative of the other half or when you recognize a good trig sub. Use integration by parts when you've got a product of two different types of functions and use the LIATE rule to figure out which function should be your $u$ vs $dv$. $\endgroup$ – user137731 Oct 5 '14 at 22:34
  • $\begingroup$ @Bye_World Ah, plain old substitution, is that it? Here I thought it was a special kind of substitution, or something. Kind of disappointing, but thanks for enlightening me. $\endgroup$ – Harald Hanche-Olsen Oct 5 '14 at 22:34
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Note that $$(x^3+x)\sqrt x=x^3\cdot x^{\frac 12}+x\cdot x^{\frac 12} =x^{\frac{7}{2}}+x^{\frac{3}{2}}$$ and that $$\int x^a dx=\frac{1}{a+1}x^{a+1}+C$$ for $a\not=-1$.

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Hint: $(x^3+x)\sqrt{x}=(x^3+x)x^{1/2}=x^{7/2}+x^{3/2}$

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