Integrating by substitution problem

[zachstrl]

Hey guys, I've spent 2 hours researching this, but to be honest I lack the mathematical vocabulary to properly research this type of integration problem. I have attached the problem below. Any help would be GREATLY appreciated. Thanks a million!

 

[tkhunny]

Hey guys, I've spent 2 hours researching this, but to be honest I lack the mathematical vocabulary to properly research this type of integration problem. I have attached the problem below. Any help would be GREATLY appreciated. Thanks a million!

What antiderivative did you manage? (Before applying limits)

 

[HallsofIvy]

I would think that u= 8x+ 3 would be a pretty obvious substitution. With that, the integrand becomes \(\displaystyle y= \frac{1}{\sqrt{u}}= u^{-1/2}\) and \(\displaystyle du= 8dx\) so that \(\displaystyle dx= \frac{1}{8}du\). The integral is from m= 0 to x= a. When x= 0, u= 8(0)+ 3= 3 and when x= a, u= 8a+ 3. The integral becomes, in terms of u, \(\displaystyle \frac{1}{8}\int_3^{8a+ 3} u^{-1/2} du\). Can you do that?

 

[africam]

definitely \(\displaystyle u = 8x+3\) is the first substitution you must try, have you tried? (remember do \(\displaystyle 8/8.\int...\))