AP Calculus BC – Chapter 6
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AP Calculus BC – Chapter 6 Differential Equations and Mathematical Modeling 6.2: Antidifferentiation by Substitution- Day 1. Goals : Compute indefinite integrals by the method of substitution. Compute definite integrals by the method of substitution. Solve a differential equation of the form

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Goals compute indefinite integrals by the method of substitution

AP Calculus BC – Chapter 6Differential Equations and Mathematical Modeling 6.2: Antidifferentiation by Substitution- Day 1

Goals: Compute indefinite integrals by the method of substitution.

Compute definite integrals by the method of substitution.

Solve a differential equation of the form

dy/dx = f(x, y) in which the variables are separable.


Properties of indefinite integrals
Properties of Indefinite Integrals:

∫kf(x)dx = k ∫f(x)dx, any constant k.

∫(f(x)±g(x))dx = ∫f(x)dx ± ∫ g(x)dx.

Power Formulas:

∫undu = + C, n ≠ 1.

∫u-1du = ∫1/u du = ln|u| + C.



Trigonometric formulas
Trigonometric Formulas:

∫cosu du = sinu + C.

∫sinu du = -cosu + C.

∫sec²u du = tanu + C.

∫csc²u du = -cotu + C.

∫secu tanu du = secu + C.

∫cscu cotu du = -cscu + C.


Exponential and logarithmic formulas
Exponential and Logarithmic Formulas:

∫eu du = eu + C.

∫au du = au/lna + C.

∫lnu du = ulnu – u + C.

∫logau du = ∫lnu/lna du = +C.


An example
An example:

How could we evaluate ∫3x²(1 + x³)7dx?


Let s generalize
Let’s generalize:

Suppose we have functions F and G with corresponding derivatives f and g. The Chain Rule gives:

(F(G(x)))’ = F’(G(x))G’(x) = f(G(x))g(x).

Take the indefinite integral of LHS & RHS:

∫(F(G(x)))’ dx = ∫f(G(x))g(x)dx.

But, ∫(F(G(x)))’dx = F(G(x)) + C.

So, ∫f(G(x))g(x)dx = F(G(x)) + C.


Assignment
Assignment:

  • HW 6.2A: Read lesson 6.2 through example 7 on page 319 and do the following exercises:

    #1-11 (odds), 15, 17, 18, 21, 27, 30.