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Chapter 4: Problem 38

Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{0}^{1} \frac{x-1}{x+1} d x $$

Short Answer

Expert verified
The definite integral of the function (x-1)/(x+1) from 0 to 1 is equal to 1 - ln 2.

Step by step solution

01

Finding the antiderivative

First, find the antiderivative (indefinite integral) of the function \(\frac{x-1}{x+1}\). Because the function is a rational function, we can separate it into two parts and perform the integration separately. Hence, \(\int \frac{x-1}{x+1} dx = \int \left(\frac{x}{x+1} - \frac{1}{x+1} \right) dx = \int dx - \int \frac{1}{x+1} dx\). The first term is a straightforward integral, while for the second term, we can use elementary integrals to find its integral which is \( \ln |x+1| \). Therefore, the antiderivative is \(x - \ln |x + 1| + C\), with C representing the constant of integration.
02

Applying the Fundamental Theorem of Calculus

Once we have found the antiderivative, we can evaluate the definite integral from 0 to 1. This involves substituting x = 1 and x = 0 into the antiderivative and calculating the difference. \( \int_{0}^{1} \frac{x-1}{x+1} dx = (1 - \ln |1 + 1|) - (0 - \ln |0 + 1|) = 1 - \ln 2 - ln 1 = 1 - \ln 2\).
03

Verifying the result with a graphing utility

To confirm the validity of the solution, enter the function (x-1)/(x+1) into a graphical calculator and find the area under the curve from 0 to 1. The result should be approximately equal to 1 - \ln 2.

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Most popular questions from this chapter

Use the equation of the tractrix \(y=a \operatorname{sech}^{-1} \frac{x}{a}-\sqrt{a^{2}-x^{2}}, \quad a>0\) Let \(L\) be the tangent line to the tractrix at the point \(P .\) If \(L\) intersects the \(y\) -axis at the point \(Q\), show that the distance between \(P\) and \(Q\) is \(a\).

In Exercises \(27-30,\) find any relative extrema of the function. Use a graphing utility to confirm your result. \(f(x)=\sin x \sinh x-\cos x \cosh x, \quad-4 \leq x \leq 4\)

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Verify each rule by differentiating. Let \(a>0\). $$ \int \frac{d u}{u \sqrt{u^{2}-a^{2}}}=\frac{1}{a} \operatorname{arcsec} \frac{|u|}{a}+C $$

Evaluate, if possible, the integral $$\int_{0}^{2} \llbracket x \rrbracket d x$$
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