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Evaluate $$\int_1^5 \frac{x}{\sqrt{2x-1}}\mathrm dx$$ by the substitution method.

What u substitution should I use for this integral, if any, and what is the final result?

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4  
How about $u^2=2x-1$? –  J. M. Sep 23 '11 at 5:02
    
What have you tried ? –  Sasha Sep 23 '11 at 5:02

3 Answers 3

There are two substitutions that should occur to you as immediate possibilities when you look at this integral: $u = 2x-1$, and $u = \sqrt{2x-1}$. Pick one, and try it; if works, great, and if not, you can still try the other one. You don’t have to get it right on the first try, and you shouldn’t start to worry until you’ve exhausted the obvious, straightforward possibilities. In this case that won’t happen (unless you make a mistake): both substitutions work. In fact, you should try them both to see how they work out, because you’re likely to need both of these types of substitution before you’re done with your course.

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+1 for not giving the final result. –  Did Sep 23 '11 at 5:20

If $u=\sqrt{2x-1}$ then $u^2=2x-1$ so $2u\;du = 2\;dx$, and hence $u\;du = dx$.

From $u^2=2x-1$ you get $x=\text{something}$.

As $x$ goes from 1 to 5, then $u$ goes from something to something.

This is called a rationalizing substitution because it gets rid of the radical.

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try the substitution x=t+3, this will simplify your limits. And you will be able to solve it using properties of definite integrals.

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