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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. • The symbol is an integral sign, the function f is the integrand of the integral, and x is the variable of integration. Evaluating an Indefinite Integral • Evaluate 2 ( x sin x ) dx . 3 x ( x sin x)dx cos x C 3 2 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.) f ( x)dx b.) f (u )du c.) f (u )dx 4 x 3 a.) f ( x)dx ( x 1)dx x C 4 4 u b.) f (u )du (u 1)du u C 4 2 4 8 (x ) x 2 x C x2 C 4 4 3 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.) f ( x)dx b.) f (u )du c.) f (u )dx f (u )dx (u 3 1)dx (( x2 )3 1)dx 7 x ( x 1)dx x C 7 6 Using Substitution • Evaluate • Let u = cos x du/dx = -sin x du = - sin x dx Using Substitution • Evaluate • Let u = 5 + 2x³, du = 6x² dx. 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. 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. 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. 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. Evaluating a Definite Integral by Substitution • Evaluate • Let u = tan x and du = sec²x dx. That Absolute Value Again • Evaluate