Essential Knowledge 3.1A2
Essential Knowledge 3.1A2 Students will know that differentiation rules provide the foundation for finding antiderivatives.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Just like we found a bunch of rules to help us take derivatives of all the different functions, we'll need rules to help us take integrals. Fortunately, since integrals are the inverse of derivatives, all the rules are just reversed.
Note: If the dx is bothering you, just think of it as something that needs to be there. Or remember, it's the length of the base of the rectangles you are making, so it wouldn't be possible to make rectangles without it.
Note: If the +C is bothering you, just think of it as something that needs to be there. Or remember, an integral tells you how much a value has changed, but you'll need a C to know where it started.
Power Functions
Since the differentiation rule was
The integration rule is
You'll use this rule the most. It's easy to get mixed up with these. I usually just check by differentiating the answer, usually something you can do in your head.
Example
checking
Taking the derivative gets us right back to x4, so we must have done it right
Example
checking
Coefficient Rule
You can think of this as just a special case of the Power Functions rule. In this case f(x) = c =cx0
Or just remember, the integral of dx is x.
Polynomial Functions
Just use the power rule on each term
Rational Functions
A rational function is a polynomial divided by a polynomial. For derivatives, we had a rule called the "quotient rule". In integration, there is no nice rule like this, and that's a big problem! When you have division in a integral, you'll have to learn rules for different cases, some of which are pretty unexpected.
Exponential Functions
The rule for derivatives was
Therefore the rule for integration is
Notice something special would happen for bases of e. (And the base will almost always be e, by the way.)
Logarithmic Functions
The derivative rule is
Therefore the rule for integration is
Notice something special would happen for bases of e. (And the base will almost always be e, by the way.) Remember, logs with base e are called "natural logs" and change to "ln". In other words, remember that ln(x) = loge(x)
yolo
Trigonometric Functions
Sine, cosine, tangent, cosecant, secant, and cotangent all have antiderivatives. You should memorize them, especially sin(x) and cos(x)
Inverse Trigonometric Functions
Inverse sine, inverse cosine, inverse tangent, inverse cosecant, inverse secant, and inverse cotangent all have integrals. Unfortunately you'll have to memorize some of these, but you'll see which ones later.
!!!!!!!!!!!!After this, go straight to 3.2C1. Might not be the best way to teach but it just makes more sense to me so let's try it