1. The function f(x) has the derivative f '(x) = n, where n is a constant. Which of the following integralexpressions is equivalent to f(x)] dx? Show your work for credit.A. Sn[ftx) ]*+cB. S[Sx)]*+cc. [*)]*+cD. [Mx)]°+cE. [Sx)]°+cF. [*) ]*+c6nG. None of the above

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Description: 1. The function f(x) has the derivative f '(x) = n, where n is a constant. Which of the following integralexpressions is equivalent to f(x)] dx? Show your work for credit.A. Sn[ftx) ]*+cB. S[Sx)]*+cc. [*)]*+cD. [Mx)]°+cE. [Sx)]°+cF. [*) ]*+c6nG. Non

Answered: 1. The function f(x) has the derivative…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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1. The function f(x) has the derivative f '(x) = n, where n is a constant. Which of the following integral
expressions is equivalent to f(x)] dx? Show your work for credit.
A. Sn[ftx) ]*+c
B. S[Sx)]*+c
c. [*)]*+c
D. [Mx)]°+c
E. [Sx)]°+c
F. [*) ]*+c
6n
G. None of the above

Expert Solution

Step 1

Given, for a function $f (x)$ , the derivative is $f' (x) = n, n$ is the constant.

To find integral $\int {[f (x)]}^{5} d x$ substitute $u = f (x)$ .

$\Rightarrow d u = f' (x) d x$

Since, $f' (x) = n$ , $\Rightarrow d u = n d x$ $u = f (x)$ .

$\Rightarrow d x = \frac{1}{n} d u$ .

Substitute $u = f (x), d x = \frac{1}{n} d u$ in integral $\int {[f (x)]}^{5} d x$ .

$\begin{array}{rcl} \int {[f (x)]}^{5} d x & = & \int u^{5} (\frac{1}{n} d u) \\ = & \frac{1}{n} \int u^{5} d u \end{array}$

Use power rule of integration, $\frac{d}{d x} (x^{n}) = \frac{x^{n + 1}}{n + 1}$

$\begin{array}{rcl} \int \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} [f (x)] \end{array} \begin{array}{rcl} \end{array}$ ⁵ dx=1n·u66+C ∫ fx⁵ dx=16nu⁶+C

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