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Chapter 4: Problem 39

Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{1}^{2} \frac{1-\cos \theta}{\theta-\sin \theta} d \theta $$

Short Answer

Expert verified
The definite integral equals \(\ln \left|\frac{2 - \sin(2)}{1 - \sin(1)}\right|\)

Step by step solution

01

Identify the Substitution

Observe the integrand and try to identify a function \(u(\theta)\) and its derivative \(du/d\theta\). Here, choose \(u(\theta) = \theta - \sin(\theta)\) because its derivative, which is \(du/d\theta = 1-\cos(\theta)\), is present in the integrand.
02

Substitute and Simplify

Express the integrand in terms of \(u\). Replace \(\theta - \sin(\theta)\) by \(u\) and \(1 - \cos(\theta) d\theta\) by \(du\). The integrand simplifies to \(1/u\). Simultaneously, transform the limits of integration according to the chosen \(u(\theta)\). When \(\theta = 1\), \(u = 1 - \sin(1)\) and when \(\theta = 2\), \(u = 2 - \sin(2)\). Therefore, the integral turns into: \(\int_{1 - \sin(1)}^{2 - \sin(2)} du / u\).
03

Solve the Integral

The integral of \(1/u\) with respect to \(u\) is \(\ln |u|\), so the antiderivative of \(du/u\) is \(\ln |u|\). Therefore, we get the result as \(\ln |u| \Big|_{1 - \sin(1)}^{2 - \sin(2)}\).
04

Apply the Fundamental Theorem of Calculus

Use the Fundamental Theorem of Calculus to evaluate the definite integral. Subtract the value of the antiderivative at the lower limit of integration from the value at the upper limit. The result is \(\ln |2 - \sin(2)| - \ln |1 - \sin(1)|\), which simplifies to: \(\ln \left|\frac{2 - \sin(2)}{1 - \sin(1)}\right|\).

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Most popular questions from this chapter

Linear and Quadratic Approximations In Exercises 33 and 34 use a computer algebra system to find the linear approximation \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and the quadratic approximation \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) of the function \(f\) at \(x=a\). Use a graphing utility to graph the function and its linear and quadratic approximations. \(f(x)=\cosh x, \quad a=0\)

Find the derivative of the function. \(y=\tanh ^{-1}(\sin 2 x)\)

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

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