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Chapter 5: Q 32. (page 452)

Solve the following integral.

鈭�sec6xdx

Short Answer

Expert verified

Answer istan5x5+2tan3x3+tanx+C

Step by step solution

01

Step 1. Given information

An integral is鈭�sec6xdx

02

Step 2. Explanation

鈭�sec6xdx=鈭�sec2xsec4xdx=鈭�tan2x+12sec2xdx

Let tanx=u

鈭�sec6xdx=鈭�u2+12du=鈭�u4+2u2+1du=u55+2u33+u+C=tan5x5+2tan3x3+tanx+C

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Most popular questions from this chapter

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).

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

x2-5x+1

Describe two ways in which the long-division algorithm for polynomials is similar to the long-division algorithm for integers and then two ways in which the two algorithms are different.

True/False: Determinewhethereachofthestatementsthat follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: f(x)=x+1x-1is a proper rational function.

(b) True or False: Every improper rational function can be expressed as the sum of a polynomial and a proper rational function.

(c) True or False: After polynomial long division of p(x) by q(x), the remainder r(x) has a degree strictly less than the degree of q(x).

(d) True or False: Polynomial long division can be used to divide two polynomials of the same degree.

(e) True or False: If a rational function is improper, then polynomial long division must be applied before using the method of partial fractions.

(f) True or False: The partial-fraction decomposition of x2+1x2(x-3)is of the form Ax2+Bx-3

(g) True or False: The partial-fraction decomposition of x2+1x2(x-3)is of the form Bx+Cx2+Ax-3.

(h) True or False: Every quadratic function can be written in the formA(x-k)2+C

Solve the integral:鈭�3x+1secxdx
See all solutions

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