Explanation Given Information: The provided slope of the tangent passing through the point ( x , f ( x ) ) is x x 2 + 1 and the value of f ( 1 ) = 0 . Formula used: Follow the steps as mentioned below to solve an integral ∫ f d x by using substitution rule. Step 1 : Assume a function u in terms of x from f . Step 2 : Take the derivative d u d x of the function obtained in step 1 and solve for d x in terms of d u . Step 3 : Substitute the value of u from step 1 and the value of d x from step 2 in the provided integral and then solve the integral. The substitution rule, u = h ( x ) where h ( x ) is any function that can be substituted. The constant multiple rule, ∫ k ⋅ f ( x ) d x = k ∫ f ( x ) d x where k is a constant The special case of the power rule for the indefinite integral when n = − 1 , ∫ x − 1 d u = ln | x | + C Calculation: Consider the provided slope of the tangent passing through the point ( x , f ( x ) ) is x x 2 + 1 . The slope of a tangent is represented by the derivative of a the function f ( x ) as, f ′ ( x ) = x x 2 + 1 The function f ( x ) needs to be anti-derivative of the function f ′ ( x ) . So, compute the function f ( x ) as, f ( x ) = ∫ x x 2 + 1 d x Evaluate the integral f ( x ) = ∫ x x 2 + 1 d x by using substitution rule. Assume, a function u in terms of x from x x 2 + 1 , u = x 2 + 1 Take the derivative d u d x of the function u = x 2 + 1 , d u d x = 2 x Solve for d x , d x = 1 2 x d u Now, substitute u = x 2 + 1 and d x = d u 2 x in the integral ∫ x x 2 + 1 d x , this gives, ∫ x x 2 + 1 d x = ∫ x u ⋅ 1 2 x d u Simplify the integral by cancelling x by x

Finite Mathematics and Applied Calculus (MindTap Course List)

7th Edition

ISBN: 9781337274203

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

Videos

Question

Chapter 13.2, Problem 74E

To determine

To calculate: The value of the function f(x) if the value of f(1)=0 and the slope of the tangent passing through the point (x,f(x)) is xx2+1.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.2, Problem 73E

Chapter 13.2, Problem 75E

Chapter 13 Solutions

Finite Mathematics and Applied Calculus (MindTap Course List)

Ch. 13.1 - In Exercises 142, evaluate the integral. [HinT:...Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The value of the function f ( x ) if the value of f ( 1 ) = 0 and the slope of the tangent passing through the point ( x , f ( x ) ) is x x 2 + 1 .

Solution Summary: The author explains how to calculate the value of the function f(x) if the slope of a tangent passing through the point is

Concept explainers

Videos

To calculate: The value of the function f(x) if the value of f(1)=0 and the slope of the tangent passing through the point (x,f(x)) is xx2+1.

Want to see the full answer?

Chapter 13 Solutions