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Chapter 5: Q 38. (page 452)

Solve the following integral.

鈭�sin5xcos2xdx

Short Answer

Expert verified

Answer is-cos7x7+2cos5x5-cos3x3+C

Step by step solution

01

Step 1. Given information

An integral is鈭�sin5xcos2xdx

02

Step 2. Explanation

鈭�sin5xcos2xdx=鈭�1-cos2x2sinxcos2xdx

Let u=cosx

鈭�sin5xcos2xdx=-鈭�u6-2u4+u2du=-u77+2u55-u33+C=-cos7x7+2cos5x5-cos3x3+C

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Most popular questions from this chapter

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 given definite integral.

鈭�45鈥�1xx2+9dx

Explain why, if x=asecu, then a2sec2u鈭�a2is 鈭�atanuif x<鈭�aand is atanuif x>a. Your explanation should include a discussion of domains and absolute values.

Solve 鈭�x+2x2+4x32dxthe following two ways:

(a) with the substitution u=x2+4x;

(b) by completing the square and then applying the trigonometric substitution x + 2 = 2 sec u.

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

鈭�13-x2dx
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