Problem 91 - Calculus Early Transcendentals

Question

\(A n\) integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution \(u=\tan (x / 2)\) or equivalently \(x=2 \tan ^{-1} u .\) The following relations are used in making this change of variables. \(A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}\) $$\text { Evaluate } \int \frac{d x}{1-\cos x}$$

Step-by-Step Solution

Answer

Question: Determine the result of the integral \(\int \frac{dx}{1-\cos x}\) using the given substitution \(u=\tan\frac{x}{2}\) and relations A, B, and C. Answer: \(\int \frac{dx}{1-\cos x} = -2\left(\frac{1}{\tan\frac{x}{2}} + \tan\frac{x}{2}\right) + C\)

Step 1: Change of Variables using u-substitution

Using the given substitution, \(u = \tan(\frac{x}{2})\), or equivalently, \(x = 2 \tan^{-1} u\). Now we will find the derivative of \(x\) with respect to \(u\) and substitute the relations A, B, and C.

Step 2: Use relations A, B, and C to transform the integral

First, differentiate \(x\) with respect to \(u\), obtaining: $$\frac{dx}{du} = \frac{d}{du}(2 \tan^{-1} u) = \frac{2}{1+u^2}$$ Now, we can apply Relation A to obtain \(dx\): $$dx = \frac{2}{1 + u^2}du$$ We will use Relation C to rewrite \(\cos x\): $$\cos x = \frac{1 - u^2}{1 + u^2}$$ Now, transform the given integral with substitutions: $$\int\frac{dx}{1-\cos x} = \int\frac{\frac{2}{1+u^2}}{1-\frac{1-u^2}{1+u^2}}du$$

Step 3: Simplify the integral

Now we will simplify the integral expression: $$\int\frac{\frac{2}{1+u^2}}{1-\frac{1-u^2}{1+u^2}}du = \int\frac{2}{\frac{(1-u^2)-1}{1+u^2}}du = \int\frac{2(1+u^2)}{-u^2}du = \int\frac{-2(1+u^2)}{u^2}du$$

Step 4: Integrate the rational function

We will now integrate the simplified expression: $$\int\frac{-2(1+u^2)}{u^2}du = -2\int\frac{1+u^2}{u^2}du= -2\int\left(\frac{1}{u^2}+1\right)du = -2\left(\int\frac{1}{u^2}du + \int du\right)$$ Now integrate the two separate terms: $$-2\left(\int\frac{1}{u^2}du + \int du\right) = -2\left(-\frac{1}{u} + u\right) + C$$

Step 5: Convert back to the original variable x

Finally, we need to convert the result back to the original variable \(x\). Recall that \(u = \tan\frac{x}{2}\), and therefore: $$-\frac{2}{u} + 2u = -2\left(\frac{1}{\tan\frac{x}{2}} + \tan\frac{x}{2}\right) + C$$ So, the final result of our integral is: $$\int \frac{dx}{1-\cos x} = -2\left(\frac{1}{\tan\frac{x}{2}} + \tan\frac{x}{2}\right) + C$$

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a powerful technique often used to solve integrals that involve trigonometric functions. When faced with complex trigonometric expressions, this method allows us to simplify the integral by introducing another variable. This is particularly useful when the original integral is challenging to integrate directly.

In our exercise, we used the substitution \(u = \tan(\frac{x}{2})\) to transform the integral. This change of variable helps convert the trigonometric functions into a form that is easier to manage. The identities related to this substitution, such as \(\sin x\) and \(\cos x\) expressed in terms of \(u\), are crucial for simplifying the integral. These substitutions are derived from standard identities and play a key role in converting the problem into one that involves rational functions.

Rational Integrands

Rational integrands are expressions that are ratios of polynomial functions. Once we transform an integral using trigonometric substitution, the resulting expression often becomes a rational integrand.

For example, in the exercise provided, after using the trigonometric identities and substitutions, we simplified the integral into a rational form: \(\int\frac{-2(1+u^2)}{u^2}du\).

This form makes it much easier to perform the integration since polynomial terms are straightforward to integrate. The simplification process involves transforming the trigonometric expressions into polynomial terms, turning the problem into one that is typically easier to handle.

Calculating Integrals

Calculating integrals is a fundamental skill in calculus. It involves finding the antiderivative or the indefinite integral of a function. In our scenario, once the trigonometric substitution has turned our integral into a rational integrand, we proceed to calculate the integral.

This process may include simplifying the integrand, such as breaking it into easier pieces that can be separately integrated. For instance, in our example, the term \(-2\left(\frac{1}{u^2}+1\right)\) was separated into two simpler integrals: \(-2\int\frac{1}{u^2}du\) and \(-2\int du\).

Each terms was integrated individually to find the antiderivative, which were then summed. Calculating integrals may also require careful algebraic manipulation to ensure the terms are in the simplest form possible before performing the integration.

U-Substitution

U-substitution is a common and useful strategy in integration. It involves substituting part of the integrand with a new variable, often denoted as \(u\), to simplify the integration process.

In this exercise, after substituting \(u = \tan(\frac{x}{2})\), we utilized differentiation to find \(dx\) in terms of \(du\). The expression \(dx = \frac{2}{1+u^2}du\) was derived, which replaced \(dx\) in the integral. This is a direct application of u-substitution, aiming to simplify the integrand to one that is easier to integrate.

By carefully choosing \(u\) and writing \(dx\) in terms of \(du\), we effectively transformed a complex integral into a simpler one. U-substitution often goes hand-in-hand with other integration techniques, like partial fraction decomposition or trigonometric substitutions, to solve challenging integrals.