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Chapter 5: Q. 51 (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�2鈭�exexdx

Short Answer

Expert verified

The solution of the given integral is 鈭�2鈭�exexdx=-2e-x鈭�x+C.

Step by step solution

01

Step 1. Given Information 

Solving the given integrals.

鈭�2鈭�exexdx

02

Step 2. Firstly simplifying the integral.

鈭�2鈭�exexdx=鈭�2ex鈭�exexdx鈭�2鈭�exexdx=鈭�2e-x鈭�1dx鈭�2鈭�exexdx=2鈭�e-xdx鈭�鈭�1dx鈭�2鈭�exexdx=-2e-x鈭�x+C

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Most popular questions from this chapter

Explain why using trigonometric substitution with x=atanuoften involves a triangle with side lengths a and x and hypotenuse of lengthx2+a2

Solve the integral:鈭�lnx3dx.

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

Consider the integral 鈭�sinxcosxdx.

(a) Solve this integral by using u-substitution with u=sinx and du=cosxdx.

(b) Solve the integral another way, using u-substitution with u=cosx and du=鈭�sinxdx.

(c) How must your two answers be related? Use algebra to prove this relationship.

Find three integrals in Exercises 39鈥�74 that can be solved by using a trigonometric substitution of the form x=atanu.
See all solutions

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