Evaluating Integral with Change of Variables

[karush]

Use change of variables to evaluate the following integral
$$\int_{2}^{7}\frac{x}{\sqrt[3] {x^2 -2 }} \,dx\approx 8.577$$
$u={x}^{2}-2$
$du=2x\ dx$
$\frac{1}{2}du=x\ dx$
$u(2)=2$ $u(7)=47 $
$$\frac{1 }{2 }\int_{2 }^{47 }\frac{1 }{ \sqrt[3] {u }} \,du $$

Can't get same answer

 

[Jameson]

Hi karush,

Your substitution looks good to me! I think your error must be in evaluating the transformed integral. Can you show your work?

 

[karush]

Before substuting back

$$\frac{1}{2}\left[\frac{3{u}^{2/3}}{2}\right]+C$$

 

[Jameson]

Yep that's correct! You can either substitute back and plug in the $x$ values or just plug in the $u$ values into what you have now. Both should give the same answer. What do you get when you try to evaluate the bounds?

 

[karush]

if
$$g(u)=\frac{1}{2}\left[\frac{3{u}^{2/3}}{2}\right]$$
then
$$g\left(47\right)-g(2)\approx8.57$$

dont think we need the +C if it is a definite integral

 

[suluclac]

Notice that the lower integral limit stays as even prime.
Also, I hope you got \(\displaystyle \frac{3}{4}(\sqrt[3]{2209}-\sqrt[3]{4})\approx8.6\).