(Basic) Please give me advice on my u-substitution technique

  • Thread starter 939
  • Start date
In summary, the two equations that the student tried to solve had different results because of the different exponents that were used. The first equation had an exponent of 2, and the second equation had an exponent of -2. The anti-derivative for the first equation was found by dividing the initial guess by the multiplicative constant, while the anti-derivative for the second equation was found by writing down the derivative of the initial guess.
  • #1
939
111
2

Homework Statement



I learned integration, but I think somewhere my technique is flawed, because I consistently get certain questions right and certain questions wrong. I don't know why I started doing it this way, but I always find DX by dividing it by DU and substitute it back into the equation. Where can I improve and what am I doing wrong?

Homework Equations



Here are two equations. First one I get right, second one wrong.
1) integral of (1 - 2x)^4 2dx
2) integral of ((5)(3t - 6)^2

The Attempt at a Solution



1)
U = 1 - 2x
DU = -2 dx
DX = DU/-2

(U)^4 (2)(DU/-2)
U^4 (-1)
-1 integral of (1/5) U^5 + C
= -1/5 (1-2x)^5 + c
RIGHT


2)
U = 3t - 6
DU = 3 dx
dx = (du/3)

5(u)^-2 du/3

1/3 5(u)^-1
(1/3) (-5/1) (3t - 6)^-1
WRONG


?
 
Last edited:
Physics news on Phys.org
  • #2
integral of ((5)(3t - 6)^2
... too many brackets. From your later statements I'm guessing you want:

$$\int 5(3t-6)^{-2} dt$$

However: Your substitution seems fine - I think you need to check how to integrate negative powers.
 
  • #3
Is the exponent 2 or -2?
 
  • #4
Wait:
(1/3) (-5/1) (3t - 6)^-1
... looks here like the power has been integrated properly (if it is -2). There is a minus sign missing in the previous line.
 
  • #5
I think that u-substitution is a technique that is used at times when it really isn't necessary. IMO, u-substitution is a good way to program a computer to do an integration, but it is usually not a good way to integrate (well, really we do u-substitution to find an anti-derivative, which we use to do integration.) So, how would I do this problem?

Let's try to find the anti-derivative of[itex]2(1-2x)^4[/itex]. Now, we have something raised to the power of 4, and so a good guess at an anti-derivative would be to just write down:
[itex](1/5)(1-2x)^5[/itex]. Now, we want to see if this is correct (ie is the derivative of our first guess what we started out with.) Well, the derivative is [itex]-2(1-2x)^4[/itex]. This is VERY close to what we started out with, in fact, it is a multiplicative constant away (what constant?) So, just divide our initial guess by this multiplicative constant and check it again, and we see that this is the correct anti-derivative.

Now, if you try to do it like this, after a while, it will become second nature. It might have seen confusing the way I described it, but I think if you write it out, it will make sense. The fact of the matter is that u-substitution can sometimes make things MUCH simpler, but, IMO, these cases are few and far between. So, use u-sub when you HAVE to, but it should be a last resort (and after practice, you will realize which integrals really "need" u-sub.)
 

1. What is the purpose of u-substitution in calculus?

The purpose of u-substitution is to simplify and solve integrals by replacing a complex expression with a simpler one, often using a variable substitution.

2. How do I know when to use u-substitution?

U-substitution is typically used when the integrand contains a product of two functions, one of which is the derivative of the other. This is known as the chain rule and is a common indicator for the use of u-substitution.

3. What are the steps for u-substitution?

The steps for u-substitution are as follows: 1) Identify a function and its derivative in the integrand, 2) Let u be the function, 3) Rewrite the integral in terms of u, 4) Find the derivative of u, 5) Substitute the integral in terms of u, and 6) Solve the new integral in terms of u.

4. Can u-substitution be used for all integrals?

No, u-substitution is not always applicable and there are other integration techniques that may need to be used depending on the complexity of the integral.

5. What are some common mistakes to avoid when using u-substitution?

Some common mistakes to avoid when using u-substitution include forgetting to change the limits of integration, not substituting all terms in the integral, and not using the chain rule when finding the derivative of u.

Similar threads

  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
Replies
27
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
719
  • Calculus and Beyond Homework Help
Replies
22
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
742
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
10
Views
282
Back
Top