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**6.2 Antidifferentiation by Substitution**

If y = f(x) we can denote the derivative of f by either dy/dx or f’(x). What can we use to denote the antiderivative of f? We have seen that the general solution to the differential equation dy/dx = f(x) actually consists of an infinite family of functions of the form F(x) + C, where F’(x) = f(x). Both the name for this family of functions and the symbol we use to denote it are closely related to the definite integral because of the Fundamental Theorem of Calculus.

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**The symbol is an integral sign, the function f is**

the integrand of the integral, and x is the variable of integration.

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**Evaluating an Indefinite Integral**

Evaluate

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**Paying Attention to the Differential**

Let f(x) = x³ + 1 and let u = x². Find each of the following antiderivatives in terms of x: a.) b.) c.)

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**Paying Attention to the Differential**

Let f(x) = x³ + 1 and let u = x². Find each of the following antiderivatives in terms of x: a.) b.) c.)

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**Using Substitution Evaluate Let u = cos x du/dx = -sin x**

du = - sin x dx

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Using Substitution Evaluate Let u = 5 + 2x³, du = 6x² dx.

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**Using Substitution Evaluate**

We do not recall a function whose derivative is cot 7x, but a basic trigonometric identity changes the integrand into a form that invites the substitution u = sin 7x, du = 7 cos 7x dx.

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**Setting Up a Substitution with a Trigonometric Identity**

Find the indefinite integrals. In each case you can use a trigonometric identity to set up a substitution.

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**Setting Up a Substitution with a Trigonometric Identity**

Find the indefinite integrals. In each case you can use a trigonometric identity to set up a substitution.

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**Setting Up a Substitution with a Trigonometric Identity**

Find the indefinite integrals. In each case you can use a trigonometric identity to set up a substitution.

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**Evaluating a Definite Integral by Substitution**

Evaluate Let u = tan x and du = sec²x dx.

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**That Absolute Value Again**

Evaluate

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Homework!!!!! Textbook – p # 1 – 6, 18 – 42 even, 54 – 66 even.

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5.3 Definite Integrals and Antiderivatives. What you’ll learn about Properties of Definite Integrals Average Value of a Function Mean Value Theorem for.

5.3 Definite Integrals and Antiderivatives. What you’ll learn about Properties of Definite Integrals Average Value of a Function Mean Value Theorem for.

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