Substitution in Indefinite Integrals

In other words…the Chain Rule for Antiderivatives

Recall: Chain Rule for Derivatives

We knew to use the Chain Rule for Derivatives if we had a composition of two (or more) functions.

If we were asked to find the derivative of the following functions, we would use the Chain Rule for Derivatives:

sin 3x

2cos 5x

42 3x

3 5x

3cos 3x

10cos 5 sin 5x x 2cos 5x

1/ 23 5x

38 2 3x

3

2 3 5x

Chain Rule for Antiderivatives

It stands to reason that the Chain Rule for Derivatives has a brother!

Unfortunately, this guy is sometimes hard to recognize.

Luckily, trig functions, square roots, and parentheses are also triggers for the Chain Rule for Antiderivatives.

We should also be on the lookout for functions and their own derivative in an integrand.

Let’s look at some examples that will help us learn what to look for.

Can We Find the Antiderivative?

2cos x dx 2sin x CYES!

xe dx YES!xe C

3dxx YES!

YES!

3ln x C

3/ 22

3x Cxdx

cos 2x dx NOPE! But it can’t be THAT hard! Right???

Can We Find the Antiderivative?

cos 2x dx What prevents us from going ahead and finding

the antiderivative?

The (2x) inside the function…of course!

Now that we’ve identified the problem, can you tell me WHY it’s a problem?

Because of the Chain Rule…of course!

The very thing that gives us the problem is where we’ll find our solution.

Wow…that’s deep.

Let’s Give it a Guess

cos 2x dx There is a pattern and even a technique to

antidifferentiate composite functions.

To discover the pattern, let’s first make a guess at what the antiderivative looks like.

Since we have a cosine function, it makes sense that it’s antiderivative must include the sine function.

Since we are not allowed to change the argument (the 2x), a good guess for our antiderivative would be…

sin(2x) + C…very good!

Let’s Check Our Guess

Our Guess: sin 2x C How can we check our guess at the antiderivative

of cos(2x)?

Take the derivative…of course!

If the derivative of sin(2x) equaled cos(2x) we would be correct in our guess…but it doesn’t!

The derivative of sin(2x) = 2cos(2x).

How does our “answer” differ from our original integrand?

Exactly…our “answer” differs by a constant of 2.

Now…Let’s FIX Our Guess

Our Guess: sin 2x C Okay…so we’ve discovered that our “answer” is

off by a constant of 2.

When your “answer” is off by a constant, this is an EASY problem to fix.

What could we multiply 2 by so that it becomes one?

1/2…of course!

Now we know what our fix is, so let’s do it!

1/2 to the Rescue!

Remember that…

didn’t quite equal

Since we decided we needed a 1/2 to fix our guess…our new guess is…

Check our new guess by differentiating.

The 2 that comes out from the Chain Rule is fixed by the 1/2. We are correct.

So…

cos 2x dx sin 2x C

1sin 2

2x C

cos 2x dx 1sin 2

2x C

Let’s Try Another One…

Make a guess at the antiderivative…

Check your guess by differentiating and see what pops out b/c of the Chain Rule…

So…we need to “fix” our guess with a 1/3.

We now believe that…

2sec 3x dx tan 3x C

23sec 3x

2sec 3x dx 1tan 3

3x C

A Couple More…

Make a guess at the antiderivative…

“Fix” your guess…

Check by differentiating…

Make a guess at the antiderivative…

See what pops out b/c of the CR…

“Fix” your guess…

sec 5 tan 5x x dx sec 5x C

1sec 5

5x C

sin 4x dx cos 4x C

1cos 4

4x C

Okay, Ms. Young…What if I Don’t Get It?

Sometimes it’s hard to just “see” what the antiderivative is when you have composite functions.

Also, we will be dealing with nastier functions that are VERY difficult to guess and check.

So…if you don’t like this guessing and checking stuff, there is a technique that we can use that will help us.

It’s called…u-Substitution…or u-Sub for the cool kids!

But I like to think of it as the Chain Rule for Antiderivatives.

U-Substitution

The “inside” becomes a “u”…

Find du/dx …

Move the dx to the other side…

Solve for dx…

Replace 2x with u and dx with du/2

Rewrite so you can find the antiderivative…

cos 2x dx2u x

2du

dx

cos2

duu

1cos

2u du

2du dx

2

dudx

1sin

2u c 1

sin 22

x c

U-Substitution…another example!

The “inside” becomes a “u”…

Find du/dx …

Move the dx to the other side…

Solve for dx…

Replace the “inside” with u and dx

Rewrite so you can find the antiderivative…

sin 3 1x dx3 1u x

3du

dx

sin3

duu

1sin

3u du

3du dx

3

dudx

1cos

3u c 1

cos 3 13

x c

U-Substitution…another example!

The “inside” becomes a “u”…

Find du/dx …

Move the dx to the other side…

Solve for dx…

Replace the “inside” with u and dx

Rewrite so you can find the antiderivative…

6 5x dx6 5u x

6du

dx

6

duu

1/ 21

6u du

6du dx

6

dudx

3/ 21 2

6 3u c

16 5

9x c