Substitution in Indefinite Integrals

Substitution in Indefinite Integrals In other words…the Chain Rule for Antiderivatives

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: Recall: Chain Rule for Derivatives

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. Chain Rule for Antiderivatives

YES! YES! Can We Find the Antiderivative? YES! YES! NOPE! But it can’t be THAT hard! Right??? 

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. Can We Find the Antiderivative?

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 Give it a Guess

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. Let’s Check Our Guess

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! Now…Let’s FIX Our Guess

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… 1/2 to the Rescue!

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… Let’s Try Another One…

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… A Couple More…

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

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… U-Substitution

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… U-Substitution…another example!

Substitution in Indefinite Integrals