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 $f$ and $g$ are continuous functions and that $k$ is any real number.
$\int \frac{f' (x) g (x) - f (x) g' (x)}{(g {(x))}^{2}} d x =____$

Short Answer

Expert verified

$\int \frac{f' (x) g (x) - f (x) g' (x)}{(g {(x))}^{2}} d x = (\frac{f (x)}{g (x)}) + C$

Step by step solution

Step 1. Given Information

$\int \frac{f' (x) g (x) - f (x) g' (x)}{(g {(x))}^{2}} d x =____$

Step 2. Solving the expression

As quotient rule of derivative is $\frac{d}{d x} (\frac{u}{v}) = \frac{u' v - u v'}{v^{2}}$

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

\int d (\frac{f (x)}{g (x)}) = \int (\frac{f (x)' g (x) - f (x) g' (x)}{(g {(x))}^{2}}) d x

\int \frac{f (x)' g (x) - f (x) g' (x)}{(g {(x))}^{2}} d x = (\frac{f (x)}{g (x)}) + C

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Indefinite integrals of combinations: Fill in the blanks to complete the integration rules that follow. You may assume that $f$ and $g$ are continuous functions and that $k$ is any real number.
$\int \frac{f' (x) g (x) - f (x) g' (x)}{(g {(x))}^{2}} d x =____$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solving the expression

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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.∫f'(x)g(x)−f(x)g'(x)(g(x))2dx=____

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solving the expression

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Mechanics Maths

Logic and Functions

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.

Indefinite integrals of combinations: Fill in the blanks to complete the integration rules that follow. You may assume that $f$ and $g$ are continuous functions and that $k$ is any real number.
$\int \frac{f' (x) g (x) - f (x) g' (x)}{(g {(x))}^{2}} d x =____$