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Chapter 4: Q. 33 (page 362)

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.

∫sec(3x)tan(3x)dx

Short Answer

Expert verified

The solution of the integral is13sec3x+C.

Step by step solution

01

Step 1. Given Information.

The given integral is∫sec(3x)tan(3x)dx.

02

Step 2. Solve. 

By solving the integral we get,

∫sec(3x)tan(3x)dxUse∫sec(x)tan(x)dx=secx+C=13sec3x+C

03

Step 3. Verification. 

To verify the answer we differentiate 13sec3x+Cit.

On differentiating we get,

13sec3x+C=ddx13sec3x+ddxC=313sec3xtan3x+0=sec3xtan3x

Hence proved.

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Most popular questions from this chapter

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].

Determine which of the limit of sums in Exercises 47–52 are infinite and which are finite. For each limit of sums that is finite, compute its value

limn→∞∑k=1nk3n4+n+1

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

∫-2-2x(f(x)+3)2dx

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

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