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Chapter 4: Q. 53 (page 404)

Indefinite integrals of combinations: Fill in the blanks to complete the integration rules that follow. You may assume that fand gare continuous functions and thatk is any real number.
鈭�f'(x)g(x)鈭�f(x)g'(x)(g(x))2dx=____

Short Answer

Expert verified

鈭�f'(x)g(x)鈭�f(x)g'(x)(g(x))2dx=f(x)g(x)+C

Step by step solution

01

Step 1. Given Information

鈭�f'(x)g(x)鈭�f(x)g'(x)(g(x))2dx=____

02

Step 2. Solving the expression

As quotient rule of derivative isddxuv=u'v-uv'v2

So,

role="math" localid="1649792911630" width="242" height="47">ddxf(x)g(x)=f(x)'g(x)-f(x)g'(x)(g(x))2
role="math" localid="1649792902040" width="258" height="47">df(x)g(x)=f(x)'g(x)-f(x)g'(x)(g(x))2dx

Integrating both sides

鈭�df(x)g(x)=鈭�f(x)'g(x)-f(x)g'(x)(g(x))2dx
鈭�f(x)'g(x)鈭�f(x)g'(x)(g(x))2dx=f(x)g(x)+C

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Most popular questions from this chapter

Repeat Exercise 13 for the function f shown above at the right, on the interval [-2,2]

Prove Theorem 4.13(c): For any real numbers a and b, 鈭�abx2dx=13b3-a3.Use the proof of Theorem 4.13(a) as a guide.

Find the sum or quantity without completely expanding or calculating any sums.

Given 鈭�k=14ak=7and,鈭�k=04bk=10,a0=2. Find the value of鈭�k=042ak+3bk.

Given a simple proof that鈭�k=0n3ak=3鈭�k=0nak

Explain why the formula for the integral of xkdoes not

apply whenk=-1. What is the integral of x-1?
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