ggcheck said:I used u substitution to get to this point:
[tex]\lim_{R\rightarrow\ 3-}} \int_{-3}^{R-3} (\frac{u + 3}{sqrt{u}}) du[/tex]
is the only way to proceed from here using integration by parts?
ahhh, I didn't think to break up the fraction, for some reason I missed thatsutupidmath said:If your calculations are correct up to this point, i do not think you need to use integ by parts. Simply just try to rearrange the integrand like this
(u+3)/u^1/2= u/u^1/2 +3/u^1/2 = u^1-1/2 +3u^-1/2 and i guess you will be fine!
Integration by parts is a technique used to evaluate integrals that involve a product of two functions. By breaking down the integral into smaller parts and applying the integration by parts formula, it is possible to simplify the integral and find a solution. This method is particularly useful for improper integrals, which do not have finite limits of integration.
The integration by parts formula is:
∫ u(x)v(x) dx = u(x) ∫ v(x) dx - ∫ u'(x) ∫ v(x) dx
where u(x) and v(x) are two functions, and u'(x) is the derivative of u(x).
When choosing u(x) and v(x), it is important to consider which function will simplify after taking the derivative and which function will become simpler after being integrated. Generally, the function that becomes simpler after being differentiated should be chosen as u(x) and the function that becomes simpler after being integrated should be chosen as v(x).
No, integration by parts is most effective for integrals that involve a product of two functions. It may not work for other types of integrals, such as trigonometric or exponential functions, where other integration techniques may be more suitable.
Yes, integration by parts may not always result in a solution. It is important to check if the integral can be solved using other techniques as well. Additionally, integration by parts may not work for integrals with infinitely oscillating functions or those that do not converge.