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Unformatted text preview: Integral Calculus & Ord. Diff. Equations Fall 2017-18 Chapter 1 Integration 1.1 Indefinite Integrals Indefinite integration may be regarded as the inverse operation to differentiation. This means that the derivative of an indefinite integral of a function is the function itself. Definition: Suppose F ( x) f ( x) . Then F (x ) is said to be an indefinite integral or anti-derivative of f (x) . This is written as ò f (x)dx = F(x) + C Note that, if C is any constant, the derivative of the right hand side is f (x) , since the derivative of a constant is . Such a constant is called the constant of integration. The function f (x) is called the integrand. Examples: d x x4 C x3 x dx C , since dx 4 4 cos ax d cos ax C , since sin ax dx C sin ax dx a a 4 3 For ease of reference, Table below gives the indefinite integrals of some of the commonly occurring functions whose validity can be established by taking the derivative of the right hand side. Table of Elementary Integrals n 1 x C n 1 1. x 4. sin ax dx a cos ax C 6. n dx (n 1) 1 1 a x 2 2 dx sin 1 x C a 2. x dx ln x C 1 1 a 3. e ax dx e ax C 1 a 5. cos ax dx sin ax C 7. a 2 1 1 x dx tan 1 C 2 x a a Here a is any constant not 0 and C is a constant of integration. 1 x 1 . The domain of ln x is x > 0 while the domain of x is real numbers with x 1 x 0 . The integration dx includes positive and negative values of x, with x 0 . It can be verified x that if x < 0, then letting x y , y > 0, we have Note that: d ln dx d 1 1 d d ln x ln y ln y dx dy dy y x Thus for both cases, x > 0 or x < 0, we have d ln dx x 1 and hence x x dx ln x C 1 Page 1 of 13 Integral Calculus & Ord. Diff. Equations Fall 2017-18 Examples: Using table of integrals we have ∫ If ∫ = √ − = +; = sin− ∫ cos +; ∫ Thus ∫ + + = = + , tan− = + , then using chain rule of differentiation = sin + The above result may be used in evaluation of the integral directly. +; +. ≠ Examples: 1 cos(2 x 3)dx 2 sin(2x 3) C 1 2x 1 C dx sin 1 2 3 9 (2 x 1) 2 1 Quick Test 1 1. Use the elementary integration formulas to write down indefinite integrals of the following 7 (a) ∫ (e)∫ (b)∫ (f) e3 x / 2 dx (m) ∫ sin ℎ (n) ∫ √ (i) ∫ 2. (j)∫ (c)∫ cos (g) ∫ sin − (k) + x 1 (d) ∫ (h) ∫ exp − (l) ∫ √ dx − (p) ∫ (o) ∫ Write down indefinite integrals of the following: (a) ∫ (d) ∫ (g) + + cos(3x 1)dx (j) ∫ sin (b) (5 2 x) dx 3 (e) ∫ (h) √ (k) (f) ∫ + + (c) ∫ (i) dx 9 ( x 3) 2 (l) − − − e dx dx 25 4 x 3 x 2 2 . 1.2 Rules of Integration Linearity Property: Suppose f (x) and g (x) are functions whose integrals exist. Then f ( x) g ( x)dx f ( x) dx g ( x) dx k f ( x) dx k f ( x)dx (k is a constant) Page 2 of 13 Integral Calculus & Ord. Diff. Equations Fall 2017-18 This follows immediately by differentiation. Examples: 3x 4 1 1dx 3 x 4dx x 1/ 2dx dx x x 5 x1 / 2 xC 3 5 (1 / 2) 3 x5 2 x x C 5 (3cos 4x 5e 3x )dx 3 cos 4 x dx 5 e3 x dx 3 5 sin 4 x e3 x C 4 3 Exercise: 1.1 1. Write down indefinite integrals of the following: (i) (iii) (v) 2. ∫ + ∫ sin ∫ (vii) ∫ − − + cos − (iv) √ (viii) ∫ + cos − (i) (iii) ∫( v + ∫ ∫ + − 3. Find the function 4. Find the function + − (ii) ∫ +√ ) such that such that ∫ (vi) ∫ Evaluate the following integrals: (iv) ∫ = (vi) ∫ √ and = + + − ∫ (ii) = . and + + + + + + √ + + + cos cos + + sin = . 5. Find the function which satisfies the differential equation = + cos ′ and satisfy the conditions = and = . .. 6. Find the position function s(t) such that velocity = − − and = . 7. Find the charge function q(t) such that current = = . − . and = 1.3 Integration by Substitution This technique depends on the following result: Suppose G and u have continuous derivatives. Then du dx dx du du G(u( x)) C dG du dG Page 3 of 13 Integral Calculus & Ord. Diff. Equations Fall 2017-18 = Note that we can formally replace ′ by du. Examples: (a) Evaluate 3 / 2 3 1/ 2 1 x x dx . 3/ 2 Set u 1 x .Then du 32 x1 / 2 dx . 2 3 1 4 1 3 / 2 3 1/ 2 3/ 2 4 1 x x dx 3 u du 6 u C 6 1 x C Thus, (b) Evaluate x 4 x 4 dx . 2 Set, u x . Then du 2xdx . x 4 x4 Thus, An integral of the from ∫ Let, u f ( x) Then 1 1 1 du tan 1 2 2 2 2 u 4 dx u2 C 14 tan 1 x2 C 2 ′ du f ( x) du f ( x) dx dx du f ( x) dx ln u C ln f ( x) C f ( x) u Thus ∫ ′ = ln| |+ The above rule is used to evaluate the other integrals frequently. For example: 2x 3 (a) x 2 3x 5 dx ln x (b) sin x cos x dx ln sin x cos x C 2 3x 5 C cos x sin x This rule may also be used to write down the integral when the numerator differs only by multiple of a constant. Examples: (a) (b) sin 3x 1 3sin 3x 1 dx dx ln (1 cos 3x) C 1 cos 3x 3 1 cos 3x 3 sec2 2 x 1 dx ln 5 tan 2 x C 5 tan 2 x 2 Page 4 of 13 Integral Calculus & Ord. Diff. Equations Fall 2017-18 Quick Test 2 1. Write down indefinite integrals of the following: 2x 1 x2 x dx x− 1 e x dx (d)∫ dx (e) x − x+ 1 x e x Write down indefinite integrals of the following: (a) 2. 1 x 7 dx (a)∫ (b) (b)∫ + cos(x / 2) dx (d) sin( x / 2) (e)∫ + + x 2 sin x dx (f) x(1 ln x) dx 1 c s +si (c) ∫ + + 2 x cos x (c) si −c s (f) ∫ cot Exercise: 1.2 1. Evaluate the following integrals by using the indicated substitutions: (a) ∫ (c) ∫ (e) ∫ √ sec + ; +√ x ; ; = = + (b) +√ 1 sin x dx; u x x (d) ∫ sin = lnx (f) cos ; = sin x dx; u 2 x 1 x 1 2. Evaluate the following integrals. (a)∫ (d) ∫ (g) ∫ + + √ + + (b) ∫ (lnx)2 x dx (e) sin(5 / x) dx (h) ∫ x2 arctan + (c) sin x (1 cos x) 4 dx (f) ∫ (i) e x + cos(e x x ) dx Page 5 of 13 Integral Calculus & Ord. Diff. Equations Fall 2017-18 1.4 Riemann Sum Consider a function f (x) which is defined (i.e. bounded) over the closed interval [ , ]. Consider a partition P of [ , ]. into n subintervals by the points a x0 x1 x 2 x n b This partition corresponds to the subintervals [ , ], [ , ], [ , ],∙ ∙ ∙ , [ − , ] . In each [ − , choose any point such that = ∑ − = =∑ ∆ = , is called a Riemann sum for the function − ≤ ≤ . Then the sum − ∆ = on [ , ]. − − Suppose f ( x) 0 on [ a, b] . Then the Riemann sum n Sn f (c r )xr r 1 is the sum of the areas of the n rectangles shown below, and thus represents an approximation to the area under the graph on [ a, b] . Figure below illustrates the case where n = 5. Different choice of the nodal points give different values of the Riemann sums. Commonly used Riemann sums are left Riemann sum ( Riemann sum If we use = − + . = = − , right Riemann sum − + = and middle , average of the heights at end points of the subinterval, it is called the Trapezoidal Riemann sum. Page 6 of 13 Integral Calculus & Ord. Diff. Equations Fall 2017-18 Summary of Riemann Sum: is defined in the closed interval [ , ]. Let a function In evaluation of Riemann sums we commonly use equal subintervals. Dividing [ , ] into equal sub-intervals of the length − ∆ = over the interval [ , ] is Riemann sum of = ∑ = For all r if = = = ∆ =∆ ∑ = The sum will be called left Riemann sum − = right Riemann sum + − − / middle Riemann sum + Trapezoidal Riemann sum Example: The following table shows the estimated area, using different Riemann sum, under the curve . = . . = − + over the interval [ , . ] using 8 equal subintervals of length Δ = Riemann sum [ − , [ [ [ [ [ [ [ [ . . . . . . . . ∑ , , , , , , , , . . . . . . . . Δ ∗∑ left Riemann sum . . . . . . . . . . The Trapezoidal Riemann sum is, = [ + × . + . + . .× + . = . right Riemann sum . . . . . . . . . . . . . . . + . . . + . . . . . . . . . middle Riemann sum . . . . . . . . + . + . . . . . . . . . . . . Page 7 of 13 Integral Calculus & Ord. Diff. Equations Fall 2017-18 The following figures show the geometrical interpretation of the above Riemann sums, Figure 1.3. (a) Left Riemann sum, (b) Right Riemann sum, (c) Middle Riemann sum and (d) Trapezoidal sum Note that the exact value of the area is . considered later. which is calculated using the integration will be 1.5.1 Numerical Integration (The Trapezoidal Rule): The trapezoidal rule for the numerical approximation to the definite integral, maintains the similar setup as discussed in the previous section. First we subdivide the interval [ , ] into subintervals of − width = . Then on each interval we will approximate the function by a straight line joining the function values at either endpoint on the interval. The following figure illustrates the case for = . Page 8 of 13 Integral Calculus & Ord. Diff. Equations Fall 2017-18 = = Each of these shaded objects is a trapezoid (hence the rule’s name) and as we can see some of them do a very good approximation to the actual area under the corresponding segment of the curve. The area of the trapezoid in the interval [ , + ] is given by, . =( + + ) × Then sum of the area of the trapeziums (e.g. the curve and is given by, ∫ ≈( + )× +( ≈ ∆ ( + + in the above figure) will approximate the area under + +⋯+ Which is known as the composite Trapezoidal rule. )× − + ⋯+ ( + ) − + )× Example: to approximate the integral ∫ . √ + Use the Trapezoidal rule with = places. to 3 decimal Solution : Here = . , The integrand is = and = . So ∆ = =√ + . − . = . . Computing the values of the integrand starting at = . , we have . =√ + = . increasing ∆ = . and stopping at . = . . = . . =√ + . . =√ + . . =√ + = . = . Page 9 of 13 Integral Calculus & Ord. Diff. Equations . . + =√ + . Using the Trapezoidal rule, we have ∫ √ + Fall 2017-18 =√ + ≈ = . . . [ . + ≈ . . = . = . + . . + + . . Example: Evaluate ∫ √ + cos to three decimal places using Trapezoidal rule with four subintervals. [ Note that in calculating the values of cosx use radian mode] Solution Here = , = and = . So, ∆ = . = √ + cos . The integrand is We need to evaluate the integrand at the five points The values are given below: π/4 1.9254 0 2.0000 x f(x) = , , , π/2 1.7321 and . 3π/4 1.5142 π 1.4142 Using Trapezoidal rule we have ∫ √ ≈ + cos . [ + ∗ ≈ × . . ≈ . + . + . + . Exercise: 1.3 1. Estimate the value the following integrals to 3 decimal places using ‘n’ subintervals of equal length using (i) left Riemann sum, (ii) right Riemann sum, (iii) middle Riemann sum and (iv)Trapezoidal rule. (a) ∫ √ (c) ∫ exp (e) ∫− + + ( n = 4) ( n = 4) ( n = 6) (b) ∫ (d) ∫ . − + (f) ∫ sin ( n = 6) ( n = 6) ( n = 5) Page 10 of 13 Integral Calculus & Ord. Diff. Equations 2. 3. Fall 2017-18 1 0 f ( x)dx Use the following data to estimate using Trapezoidal rule. x 0 0.2 0.4 0.6 0.8 1.0 f(x) 1.000 0.82 0.652 0.527 0.434 0.368 The work done in compressing a piston is given by W 1 12 4 0 F (s)ds Use Trapezoidal rule with 4 strips to calculate W using the following tabulated results. 4. s 0 1 2 3 4 F(s) 2.7 9.1 16.4 29.90 51.2 Use the following data to estimate ∫ using Trapezoidal rule (n=6). Time (s) 0 3 6 9 12 15 18 Velocity(m/s) 4 12 18 8 20 24 22 1.5.2 Definite Integration (Riemann Integration): Let point be a bounded function defined on [ , ]. For a division of [ , ]. into n subintervals with a in each subinterval [ − , ], a Riemann sum is ∑ − = − =∑ = ∆ If the limit of the sum exists as n such that max Δ → and is independent of the choice of x r and cr , then we say that f (x) is (Riemann) integrable and the limit is written as b a f ( x) dx which is called definite integral of f (x) over [ a, b] . Thus, ∫ = lim →∞ ax ∆ → ∑ = ∆ _ The quantity a is the upper limit and b is the lower limit of the integration. Note that the above limit exists if f (x) is bounded over [ , ]. 1.6 The Fundamental Theorem of Calculus: Calculating the value of a definite integral using Riemann sums is extremely difficult for all but simple functions. However it turns out that the definite integral is related to an indefinite integral of f (x) . This is known as the Fundamental Theorem of Calculus. Page 11 of 13 Integral Calculus & Ord. Diff. Equations Fall 2017-18 Theorem: If f (x) is a continuous function on [a, b] and F (x) is an indefinite integral of f (x) then b a f ( x)dx [ F ( x)]a F (b) F (a) where ∫ = b . [For proof of this important result consult any books on Higher Calculus] For example: 2 1 (a) (3x 2 4 x 5)dx x 3 2 x 2 5 x e (b) 1 (c) 0 2 1 8 8 10 (1 2 5) 18 1 dx [ln x]1e ln e ln1 1 x /2 sin 2 x dx [ 12 cos 2 x]0 / 2 12 (1 1) 1 1.7 Some Properties of Definite Integrals: The following properties of the definite integral are stated without proof; they are extremely useful in evaluating integrals. 1. If f (x) is any integrable function then a a f ( x)dx 0 . 2. If f (x) is integrable on any interval containing three points a, b and c then b c b a f ( x)dx a f ( x)dx c f ( x)dx . 3. If f (x) is an even function, that is, if f (x) f ( x) for all x then a a f ( x)dx 2 a 0 f ( x)dx . 4. If f (x) is an odd function, that is, if f ( x) f ( x) for all x then a a f ( x)dx 0 . For example: (a) (b) (c) 2 2 (2 x 2 3)dx 2 /4 / 4 cos x dx 2 2 0 ( 2 x 2 2 3)dx since (2 x 3) is an even function. /4 / 4 cos x dx since cos x is an even function ( cos(x) cos x ). /4 / 4 sin x dx 0 since sin x is an odd function ( sin(x) sin x ). Page 12 of 13 Integral Calculus & Ord. Diff. Equations Fall 2017-18 Exercise: 1.4 1. Evaluate the following integrals. (a) ∫ (d) ∫ (g) ∫ 2. − + (b) ∫ sin + cos (e) √ − 3 cos(ln x) 1 (h) ∫ x + (c) ∫ (f) dx + (arctan x) 2 dx 1 x 2 1 0 1 0 (i) 1 dx 4 x 2 State with reason whether the following functions are odd, even or neither. = (a) + = sin (d) 3. √ + − = (b) cos (c) f ( x) sin 2 x cos3 x + (e) f ( x) x 4 e 2 x (f) f ( x) tan x cot x Evaluate the following integrals: (a) ∫− (d) ∫− / (g) ∫−/ + (b) cos sin 2 2 x(1 x cos (e) ∫− (h) ∫− 2 x 6 )dx (c) ∫− sin (f) ∫− sin (i) ∫− √ + + cos Exercise: 1.5 a. The work performed by the force b. Hooke’s Law: Solve the followings: on an object be defined by = ∫ . = , where is a constant called the spring constant. 1. A spring has a natural length of 1m. A force of 24 N stretches the spring to a length of 1.8 m. (a) Find the spring constant . (b) How much work is required to stretch the spring (c) How far will a 45 N force stretch the spring? m beyond its natural length? Page 13 of 13 ...
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