What is wrong with the following "calculation" of ∫(1)/(x^2-1) d x ? ∫(1)/(x^2-1) =∫(1)/(u)

Name: Problem 92, Chapter 13: Finite Mathematics and Applied Calculus
Uploaded: 2021-11-25T03:11:53-08:00
Duration: 1 min 37 s
Channel: Gregory Higby
Description: What is wrong with the following "calculation" of ∫(1)/(x^2-1) d x ? ∫(1)/(x^2-1) =∫(1)/(u) =ln|u|+C =ln|x^2-1|+C Using the substitution u=x^2-1

step 1 So evaluating this integral of 1 over x squared minus 1 d x. Letting you equal x squared minus 1

Question

What is wrong with the following "calculation" of $\int \frac{1}{x^{2}-1} d x ?$ $\begin{aligned} \int \frac{1}{x^{2}-1} &=\int \frac{1}{u} \\ &=\ln |u|+C \\ &=\ln \left|x^{2}-1\right|+C \end{aligned}$ Using the substitution $u=x^{2}-1$

   What is wrong with the following "calculation" of $\int \frac{1}{x^{2}-1} d x ?$
$\begin{aligned} \int \frac{1}{x^{2}-1} &=\int \frac{1}{u} \\ &=\ln |u|+C \\ &=\ln \left|x^{2}-1\right|+C \end{aligned}$
Using the substitution $u=x^{2}-1$

Finite Mathematics and Applied Calculus

Stefan Waner, Steven… 5th Edition

Chapter 13, Problem 92

Instant Answer

Solved by Expert Gregory Higby

11/25/2021

Step 1

Step 1: We start with the integral $\int \frac{1}{x^{2}-1} dx$ and we want to use the substitution $u=x^{2}-1$. Show more…

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What is wrong with the following "calculation" of $\int \frac{1}{x^{2}-1} d x ?$ $\begin{aligned} \int \frac{1}{x^{2}-1} &=\int \frac{1}{u} \\ &=\ln |u|+C \\ &=\ln \left|x^{2}-1\right|+C \end{aligned}$ Using the substitution $u=x^{2}-1$

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Key Concepts

u-Substitution

u-Substitution is a method used to simplify integration by replacing a composite part of the integrand with a single variable, u. This technique requires that after substituting, the differential du must correctly correspond to the original variable’s differential, including all necessary factors. An incorrect or incomplete substitution can lead to errors in the integral evaluation.

Chain Rule in Integration

The chain rule, a fundamental concept in differentiation, must be applied in reverse when using u-substitution in integration. This means that when a substitution is made, the derivative of the substitution expression (i.e., the factor derived from differentiating u with respect to the original variable) must be included in the differential. Neglecting this factor alters the integral’s structure and leads to incorrect results.

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