Chapter 4: Q. 54 (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.
$\int f' (g (x)) g' (x) d x =_____$

Short Answer

Expert verified

$\int f' (g (x)) g' (x) d x = f' (g (x)) + C$

Step by step solution

Step 1. Given Information

$\int f' (g (x)) g' (x) d x =_____$

Step 2. Solving the integral

Let $g (x) = t$
Differentiate with respect to ' $t$ '
$\frac{d}{d t} g (x) = \frac{d t}{d t}$

$g' (x) d x = d t$
Putting the value $\int f' (g (x)) g' (x) d x = \int f' (t) d t$

Step 3. Using the antiderivative

As $\int f (x) d x = F (x) + C$ where $F$ is the antiderivative of $f$

Here antiderivative of $f' (x) = f (x)$
$\int f' (g (x)) g' (x) d x = f (t) + C$
Putting $t = g (x)$
$\int f' (g (x)) g' (x) d x = f (g (x)) + C$

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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.
$\int f' (g (x)) g' (x) d x =_____$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solving the integral

Step 3. Using the antiderivative

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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'(g(x))g'(x)dx=_____

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solving the integral

Step 3. Using the antiderivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Statistics

Mechanics Maths

Theoretical and Mathematical Physics

Pure Maths

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.
$\int f' (g (x)) g' (x) d x =_____$