Evaluate the Definate Integral

[KEYWEST17]

This is the answer I got. Can anyone confirm.

-(564/25)+20sqrt(7)

 

[galactus]

Sorry, that's incorrect.

Perhaps try it this way. Breaking it up into easier integrations.

$\displaystyle \int_{1}^{7}\frac{8x^{2}+10}{\sqrt{x}}dx=8\int_{1}^{7}x^{\frac{3}{2}}dx+10\int_{1}^{7}x^{\frac{-1}{2}}dx$

Another way would be to make the sub $\displaystyle x=u^{2}, \;\ dx=2udu$

This results in an easy integration. Don't forget to change the integration limits, though.

 

[KEYWEST17]

Just reworked the problem using your steps and got 22512/sqrt(7)-144 and it is still incorrect. Don't know where I'm going wrong

 

[galactus]

$\displaystyle \int_{1}^{7}\frac{8x^{2}+10}{\sqrt{x}}dx$

Let $\displaystyle x=u^{2}, \;\ dx=2udu$

This changes the integration limits to $\displaystyle 1 \;\ and \;\ \sqrt{7}$

$\displaystyle \int_{1}^{\sqrt{7}}\frac{8u^{4}+10}{u}\cdot 2udu$

$\displaystyle =\int_{1}^{\sqrt{7}}\left(4(4u^{4}+5)\right)du$

\(\displaystyle =\left \frac{4u(4u^{4}+25)}{5}}\right|_{1}^{\sqrt{7}}=\frac{4(221\sqrt{7}-29)}{5}\approx 444.569\)

 

[Deleted member 4993]

or the otherway:

\(\displaystyle \int_{1}^{7}\frac{8x^{2}+10}{\sqrt{x}}dx=8\int_{1}^{7}x^{\frac{3}{2}}dx+10\int_{1}^{7}x^{\frac{-1}{2}}dx \ = \ 8*\frac{2}{5} \left [x^{\frac{5}{2}}\right ] \right|_1^7 \ + 10*2\left [x^{\frac{1}{2}}\right ] \right|_1^7 \ = \ \frac{16}{5}*[49\sqrt{7} -1] \ + \ 20*[\sqrt{7}-1]\)

and simplify......

 

[lookagain]

or the otherway:

$\displaystyle \int_{1}^{7}\frac{8x^{2} + 10}{\sqrt{x}}dx$

$\displaystyle =8\int_{1}^{7}x^{\frac{3}{2}}dx + 10\int_{1}^{7}x^{\frac{-1}{2}}dx \ =$

\(\displaystyle \ 8*\frac{2}{5} \left [x^{\frac{5}{2}}\right ] \right|_1^7 \ + 10*2\left [x^{\frac{1}{2}}\right ] \right|_1^7 \ =\)

$\displaystyle \ \frac{16}{5}*[49\sqrt{7} -1] \ + \ 20*[\sqrt{7} -1]$

and simplify......

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