Essential Knowledge 3.1A2

Essential Knowledge 3.1A2 Students will know that differentiation rules provide the foundation for finding antiderivatives.

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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) = log_e(x)

yolo

Trigonometric Functions

Sine, cosine, tangent, cosecant, secant, and cotangent all have antiderivatives. You should memorize them, especially sin(x) and cos(x)

Here they are

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.

Here they are

!!!!!!!!!!!!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