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Chapter 4: Q. 64 (page 363)

Solve each of the integrals in Exercises 63鈥�68, where a, b, and c are real numbers with a鈮�0,b鈮�0,c>1,andc鈮�0.

鈭�abxcdx.

Short Answer

Expert verified

The value of the given integral is:ax1-cb(1-c)+k.

Step by step solution

01

Step 1. Given Information.

Given is the integral:

鈭�abxcdx,a鈮�0,b鈮�0,c>1,andc鈮�0,anda,b,andcareconstants.

02

Step 2. Formula used.

鈭�xndx=xn+1n+1+k,wheren鈮�1.

03

Step 3. Solving the integral.

鈭�abxcdx=ab鈭�x-cdx=abx-c+1-c+1+k=ax1-cb(1-c)+k.

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Most popular questions from this chapter

Explain why at this point we don鈥檛 have an integration formula for the functionf(x)=secx whereas we do have an integration formula for f(x)=sinx.

If 鈭�-23f(x)dx=4,鈭�-26f(x)dx=9,鈭�-23g(x)dx=2, and 鈭�36g(x)dx=3, then find the values of each definite integral in Exercises 29-40. If there is not enough information, explain why.

鈭�-2-2x(f(x)+3)2dx

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

鈭�04(2x-3)2+5dx

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|鈭�abf(x)dx|.

(b) True or False: The area of the region between f(x) = x 鈭� 4 and g(x) = -x2on the interval [鈭�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 鈭�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].

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