Antidifferentiation by Substitution

1. 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.

2. The symbol is an integral sign, the function f is the integrand of the integral, and x is the variable of integration.

3. Evaluating an Indefinite Integral • Evaluate

5. 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.)

6. 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.)

7. Using Substitution • Evaluate • Let u = cos x du/dx = -sin x du = - sin x dx

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

9. 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.

10. 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.

11. 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.

12. 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.

13. Evaluating a Definite Integral by Substitution • Evaluate • Let u = tan x and du = sec²x dx.

14. That Absolute Value Again • Evaluate