Indefinite Integrals Consider a continuous function f.If F is an antiderivative for fon [a,bl,then (①) f(x)d=[F(x)T If C is a constant,then [F(x)+C=[F(b)+C]-[F(a)+C]=F(b)-F(a)=[F(x] Thus we can replace(1)by writing ∫f(x)dk=[F(x)+C正 If we have no particular interest in the interval [a]but wish instead to emphasize that F is an antiderivative forf,which on open intervals simply means that F=f,then we omit the a and the b and simply write ∫f(x)k=F(x)+C Antiderivatives expressed in this manner are called indefinite integrals.The constant C is called the constant of integration;it is an arbitrary constant and we can assign to it any value we choose.Each value of C gives a particular antiderivative,and each antiderivative is obtained from a particular value of C. Main Menu 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals Consider a continuous function f . If F is an antiderivative for f on [a, b], then If C is a constant, then Thus we can replace (1) by writing If we have no particular interest in the interval [a, b] but wish instead to emphasize that F is an antiderivative for f , which on open intervals simply means that F´= f , then we omit the a and the b and simply write Antiderivatives expressed in this manner are called indefinite integrals. The constant C is called the constant of integration; it is an arbitrary constant and we can assign to it any value we choose. Each value of C gives a particular antiderivative, and each antiderivative is obtained from a particular value of C. (1) ( ) ( ) b b a a f x dx F x = ( ) ( ) ( ) ( ) ( ) ( ) b b a a + = + − + = − = F x C F b C F a C F b F a F x ( ) ( ) . b b a a f x dx F x C = + f x dx F x C ( ) = + ( )
Indefinite Integrals ■Table5.6.1 sinx dx =-cosx +C cosx dx sinx +C sec2 x dx tanx +C csc2x dx =-cotx+C secx tanx dx secx +C cscx cotx dx =-cscx +C Main Meny m
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals
Indefinite Integrals The linearity properties of definite integrals also hold for indefinite integrals. (5.6.1) 「[afx+gr】dk=afex)dk+Bgx)dk. Example 1 Calculate [5x32-2csc2xdx Solution Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals The linearity properties of definite integrals also hold for indefinite integrals. Example 1 Calculate 3/ 2 2 5 2csc x x dx − Solution
Indefinite Integrals Example 2 Find fgiven that f(x)=x3 +2and f(0)=1. Main Meny 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals Example 2 Find f given that ( ) 2 and (0) 1. 3 f x = x + f =
Indefinite Integrals Example 3 Find fgiven that f"(x)=6x-2,f)=-5,andf0=3. Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals f "(x) = 6x − 2, f '(1) = −5, and f (1) = 3. Example 3 Find f given that
Indefinite Integrals Application to Motion Example 4 345678 An object moves along a coordinate line with velocity Figure 5.6.1 v(t)=2-3t+units per second Its initial position(position at time t=0)is 2 units to the right of the origin.Find the position of the object 4 seconds later. Solution Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals Application to Motion Example 4 An object moves along a coordinate line with velocity v(t) = 2 − 3t + t 2 units per second. Its initial position (position at time t = 0) is 2 units to the right of the origin. Find the position of the object 4 seconds later. Solution
Indefinite Integrals Example 5 An object moves along the x-axis with acceleration a(t)=2t-2 units per second per second.Its initial position(position at time t=0)is 5 units to the right of the origin.One second later the object is moving left at the rate of 4 units per second. (a)Find the position of the object at time t=4 seconds (b)How far does the object travel during these 4 seconds? 0 5 Figure 5.6.2 Main Meny 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals Example 5 An object moves along the x-axis with acceleration a(t) = 2t - 2 units per second per second. Its initial position (position at time t = 0) is 5 units to the right of the origin. One second later the object is moving left at the rate of 4 units per second. (a) Find the position of the object at time t= 4 seconds (b) How far does the object travel during these 4 seconds?
Indefinite Integrals Example 6 Find the equation of motion for an object that moves along a straight line with constant acceleration a from an initial position x0 with initial velocity vo. Main Meny☐ w59
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Indefinite Integrals Example 6 Find the equation of motion for an object that moves along a straight line with constant acceleration a from an initial position x0 with initial velocity v0
The u-Substitution THEOREM 5.7.1 If f is a continuous function and F=f,then f(u(x))u'(x)dx =F(u(x))+C for all functionsu=u(x)which have values in the domain of f and continuous derivative u' Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution
The u-Substitution Example 1 Calculate 「(x2-1)4dx ☒无法显示该图片 Main Menu 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The u-Substitution Example 1 Calculate ( −1) . 2 4 x dx