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Chapter 5: Q. 53 (page 418)

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

鈭�sinxcos7xdx

Short Answer

Expert verified

The solution of the given integral is 鈭�sinxcos7xdx=-18cos8x+C.

Step by step solution

01

Step 1. Given Information 

Solving the given integrals.

鈭�sinxcos7xdx

02

Step 2. Using the substitution method.

Let

u=cosxdudx=-sinxdu=-sinxdx-du=sinxdx
03

Step 3. This substitution changes the integral into 

鈭�sinxcos7xdx=-鈭�u7du鈭�sinxcos7xdx=-u7+17+1+C鈭�sinxcos7xdx=-u88+C鈭�sinxcos7xdx=-18u8+C鈭�sinxcos7xdx=-18cos8x+C

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Most popular questions from this chapter

Suppose u(x)=x2. Calculate and compare the values of the following definite integrals:

role="math" localid="1648786835678" 鈭�-15u2du,鈭�x=-1x=5u2duand鈭�u(-1)u(5)u2du

Consider the integral 鈭�1x21鈭�x2dxfrom the reading at the beginning of the section.

(a) Use the inverse trigonometric substitution u=sin鈭�1鈦�xto solve this integral.

(b) Use the trigonometric substitution x=sin鈦�u to solve the integral.

(c) Compare and contrast the two methods used in parts (a) and (b).

Solve the integral:鈭�x2e3xdx.

Explain how to know when to use the trigonometric substitutions x=asin鈦�u,x=atan鈦�u,andx=asec鈦�u, Describe the trigonometric identity and the triangle that will be needed in each case. What are the possible values for xand uin each case?

Solve the integral:鈭�ln3xdx
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