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Chapter 4: Q. 51 (page 315)

Indefinite integrals of combinations: Fill in the blanks to complete the integration rules that follow. You may assume that f and g are continuous functions and that k is any real number.

∫f(x)+g(x)dx=...

Short Answer

Expert verified

The complete integration rule is ∫f(x)+g(x)dx=∫f(x)dx+∫g(x)dx.

Step by step solution

01

Step 1. Given information. 

Consider the given indefinite integral is∫f(x)+g(x)dx.

02

Step 2. Fill the blank for the indefinite integral ∫f(x)+g(x)dx.

The theorem of sum rule for Indefinite Integrals states that, ∫f(x)+g(x)dx=∫f(x)dx+∫g(x)dx.

Therefore, it can be said the appropriate fill in black will be ∫f(x)+g(x)dx=∫f(x)dx+∫g(x)dx.

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Most popular questions from this chapter

Describe an example that illustrates that ∫ab|f(x)|dx is not equal to |∫abf(x)dx|.

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: tanx=sinxcosx).

∫2xlnx-xlnx2dx.

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The absolute area between the graph of f and the x-axis on [a, b] is equal to|∫abf(x)dx|.

(b) True or False: The area of the region between f(x) = x − 4 and g(x) = -x2on the interval [−3, 3] is negative.

(c) True or False: The signed area between the graph of f on [a, b] is always less than or equal to the absolute area on the same interval.

(d) True or False: The area between any two graphs f and g on an interval [a, b] is given by ∫ab(f(x)-g(x))dx.

(e) True or False: The average value of the function f(x) = x2-3 on [2, 6] is

f(6)+f(2)2= 33+12= 17.

(f) True or False: The average value of the function f(x) = x2-3on [2, 6] is f(6)-f(2)4= 33-14= 8.

(g) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 2] and the average value of f on [2, 5].

(h) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 3] and the average value of f on [3, 5].

Verify that∫lnxdx=x(lnx-1)+C(Do not try to solve the integral from scratch.

Your calculator should be able to approximate the area between a graph and the x-axis. Determine how to do this on your particular calculator, and then, in Exercises 21–26, use the method to approximate the signed area between the graph of each function f and the x-axis on the given interval [a, b].

f(x)=1-2x,[a,b]=[-3,1]
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