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jdawg
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Homework Statement
∫2x√(2x-3) dx
Homework Equations
The Attempt at a Solution
u=2x
du=2 dx
1/2∫u√(u-3) du
Am I on the right track with this? I'm not really sure what to do next.
Jtechguy21 said:Sort of. u= the inside of the square root
u=2x-3
du=2dx
dx=du/2 or 1/2
∫1/2 2xu^1/2 dx
the 2's cancel out.
now it should look like
∫x*u^1/2 and integrate that. and plug in the value for u after your done integrating
Becareful, because if i remember correctly. after integrating ∫x*u^1/2
Since we have an x. you have to solve for it, and plug it in.
u=2x-3 <-Solve the x
2x-3=0 2x=3
x=3/2
SteamKing said:If u = 2x -3, then 2x = u + 3, so that your integral after substitution has only u in it.
Your integrand becomes (u+3)*SQRT(u)*du/2
You don't want an integrand which mixes x and u after substitution.
"U Substitution" is a technique used in calculus to simplify integrals by substituting a new variable, usually denoted as "u", in place of the original variable. It is also known as the "change of variable" method.
"U Substitution" is typically used when the integral involves a function within a function, a composite function. It can also be used when the integrand contains a product or quotient of functions. In general, it is used to simplify complicated integrals.
To perform "U Substitution", follow these steps:
1. Identify the u-substitution by finding a function within a function, or a product or quotient of functions.
2. Choose a new variable "u" to substitute for the original variable.
3. Find the derivative of "u" with respect to the original variable.
4. Rewrite the integral in terms of "u" using the derivative found in step 3.
5. Solve the new integral in terms of "u".
6. Finally, substitute back in the original variable to get the final solution.
Using "U Substitution" can simplify complicated integrals and make them easier to solve. It also allows for the use of more basic integration techniques, such as the power rule or integration by parts. Additionally, it can help identify patterns and make connections between seemingly unrelated integrals.
Yes, there are some limitations to using "U Substitution". It may not always work for all integrals, especially those with trigonometric functions. In some cases, it may also result in a more complicated integral. It is important to carefully choose the substitution variable and check the final solution for correctness.