Integration by Substitution Exploratory Ws

8/3/2019 Integration by Substitution Exploratory Ws

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Date:_______

Integration by Substitution

Exercise 1: Weve been learning to undo the derivative with an integral. Going backwards can be a

little confusing, but youve gotten pretty good at it. But what happens when the thing that was

differentiated to begin with used the chain rule? Life gets a little more complicated. Lets look atsome chain rule differentiation problems:

1. 2.

Exercise 2: Now, look at those derivatives. Notice how the inside function appears intact and the

inside functions derivative is also present. When we take the derivative of a function using the chain

rule, the inside function remains inside, then the whole thing is multiplied by the derivative of the

inside function. Lets look at some integrals and see if you can use this pattern to figure out what theintegral equals. Please fill out the following table.

Integral Identify the

inside function

Identify the

derivative of the

inside function

Identify the

original outside

function

What do you think

the answer to the

integral is?

Double check your final answers by differentiating.

So hopefully you noticed that the key to taking these integrals is undoing the chain rule. The chain

rule produces a new component. It takes a simple and turns it into kind of a mess: . Notice that if we want to go backward, that whole component kind of goes away.So if we can correctly identify the inside function, then find its derivative in the integral were trying

to take, then undo the derivative done to the outside function, we can finesse our way back to the

original function. Theres a formal calculus way to do this finessing called u-substitution, but it does

the same things I just mentioned. It:

(1) Identifies the inside function(2)Identifies the derivative of the inside function and disappears it.(3)Undoes the derivative of the outside function to get us back to our original function.

U-substitution is just undoing the Chain rule. PLEASE dont lose sight of this, because it can become

easy to just do the u-substitution by rote and to forget that it actually makes sense.


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Exercise 3: Lets look at the following integral:

.a. Identify what you think was the original inside function and set that function equal to

b. Take the derivative of using differentials. If you dont remember how to do this, you justtake the derivative using Leibnitz notation (

) and then you multiply both sides by

c. Now, find this derivative in the integral. Because equals this derivative, just replace thatchunk of the integral with and substitute in for the part of the integral that equals.You did all the substitutions correct if there are no longer any xs.

d. You should be left with just whatever the derivative of the outside function was and youshould be able to take the integral now. Try it. (Dont forget the +C!)

e. Now, replace the in your answer with what originally equaled. Youre done. If you dontbelieve me, take the derivative of your answer and make sure get the integrand that we

started with.

Exercise 4: Try following the steps again with the following problem: a. Identify ub. Differentiate u using differentialsc. Replace the inside function with u, and the derivative of the inside function with du.d. Take the integral

e. Substitute the original inside function back in.

f. Take the derivative of the result to check your work.


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Exercise 5: Lets look at a more complicated application of u-substitution: . Well followthe same steps for as long as we can.

a. Identify the inside function and set it equal to .

b.

Differentiate using differentials.

c. Look for this derivative in the integrand. Oh no! Its not there is it? What do we do? Divideboth sides of the equality by 5. Now you should have something like

. Replace inthe integrand with

and replace the inside function with .

d. Now you can just slide that out in front of the integral symbol and integrate as normal.

e. Replace the u with the original inside function

f. Take the derivative of the result to check your work.

Exercise 6: u-substitution doesnt always work. It only works if the derivative youre trying to

integrate was generated using the chain rule- meaning theres an intact inside function and its

derivative multiplying the whole result. Try it out. Try taking the integral:


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Exercise 7: A little more practice- try taking the following integrals using integration by u-

substitution. If its not possible, just write NP.

a.

b.

(note: they like to put the in the numerator like that to be fancy. It would be

easier to take this integral, however, if you rewrote the fraction as

. Also, you

would probably want to make the whole thing in parentheses the .)

c. (Hint: Break it into 2 integrals)

d. (Hint: Make the whole thing under the radical the .)

e. (This one is really tricky. Let and replace the inside of the radical with. Youre still left with that that MUST be written in terms of u before you can integrate.How can we rewrite it in terms of ?)

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Integration by Substitution Exploratory Ws