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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. 328 # 1 – 6, 18 – 42 even, 54 – 66 even.

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