ANTIDERIVATIVES Definition: reverse operation of finding a derivative Notice that F is called AN antiderivative and not THE antiderivative. This is easily understood by looking at the example above. Some antiderivatives of 푓 푥 = 4푥3 are 퐹 푥 = 푥4, 퐹 푥 = 푥4 + 2, 퐹 푥 = 푥4 − 52 Because in each case 푑 퐹(푥) = 4푥3 푑푥 Theorem 1: If a function has more than one antiderivative, then the antiderivatives differ by a constant. • The graphs of antiderivatives are vertical translations of each other. • For example: 푓(푥) = 2푥 Find several functions that are the antiderivatives for 푓(푥) Answer: 푥2, 푥2 + 1, 푥2 + 3, 푥2 − 2, 푥2 + 푐 (푐 푖푠 푎푛푦 푟푒푎푙 푛푢푚푏푒푟) INDEFINITE INTEGRALS Let f (x) be a function. The family of all functions that are antiderivatives of f (x) is called the indefinite integral and has the symbol f (x) dx The symbol is called an integral sign, The function 푓 (푥) is called the integrand. The symbol 푑푥 indicates that anti-differentiation is performed with respect to the variable 푥. By the previous theorem, if 퐹(푥) is any antiderivative of 푓, then f (x) dx F(x) C The arbitrary constant C is called the constant of integration. Indefinite Integral Formulas and Properties Vocabulary: The indefinite integral of a function 푓(푥) is the family of all functions that are antiderivatives of 푓 (푥). It is a function 퐹(푥) whose derivative is 푓(푥). The definite integral of 푓(푥) between two limits 푎 and 푏 is the area under the curve from 푥 = 푎 to 푥 = 푏. It is a number, not a function, equal to 퐹(푏) − 퐹(푎). Example 1: 푎. 2 푑푥 = 2푥 + 퐶 푏. 16 푒푡 푑푡 = 16 푒푡 + 퐶 5 4 푥 3 5 c. 3푥 푑푥 = 3 + 퐶 = 푥 + 퐶 5 5 d. (2x5 3x2 1)dx 2x5dx 3x2dx 1dx 6 3 5 2 x x 2x dx 3x dx 1dx 2 3 1x C 6 3 1 x6 x3 x C 3 5 5 1 e. 4ex dx dx 4exdx 5 dx 4 exdx x x x 5ln x 4ex C 2 3 2 2 f . 2x 3 dx 2x 3 dx 3x4dx 2 x 3 dx 3 x4dx 4 x 5 x 3 x3 6 5 2 3 C x 3 x3 C 5 3 5 3 6 5 1 x 3 C 5 x3 x4 8x3 x3 x2 g. dx x2 8x dx 8 C 2 x 3 2 x3 4x2 C 3 4 1 1 1 6 x 3 x 2 h. 83 x dx 8 x 3dx 6 x 2 dx 8 6 C x 4 1 3 2 4 6x 3 12 x C i. (x2 2)(x 3)dx x3 3x2 2x 6dx x4 x3 x2 6x C 4 Definition: A differential equation is any equation which contains derivative(s). Solving a differential equation involves finding the original function from which the derivative came. The general solution involves C . The particular solution uses an initial condition to find the specific value of C. Differential equation is called a separable differential equation if it is possible to separate 푥 and 푦 variables. If 푑푦 = 푓(푥) 푑푥 then the process of finding the antiderivatives of each side of the above equation (called indefinite integration) will lead to the solution. 푑푦 푑푥 = 푓(푥) 푑푥 ⇒ 푦 = 퐹 푥 + 퐶 푑푥 INITIAL VALUE PROBLEMS Particular Solutions are obtained from initial conditions placed on the solution that will allow us to determine which solution that we are after. Example: 푑푦 Solve the differential equation = 3푥2 if y 2 = −3. 푑푥 Find both the general and particular solution. 푑푦 푑푥 = 3푥2 푑푥 푑푥 general solution: 푦 = 푥3 + 퐶 particular solution: y 2 = −3 ⇒ −3 = 8 + 퐶 ⇒ 퐶 = −11 푦 = 푥3 − 11 Example: Find the equation of the curve that passes through (2,6) if its slope is given by dy/dx = 3x2 at any point x. The curve that has the derivative of 3x2 is x3 2 3 3x dx 3 C x C 3 Since we know that the curve passes through (2, 6), we can find out C y x3 C 6 23 C C 2 Therefore, the equation is 푦 = 푥3 − 2 .