Problem 92 - 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. \(A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}\) $$\text { Evaluate } \int \frac{d x}{1+\sin x+\cos x}$$

Step-by-Step Solution

Answer

Question: Evaluate the integral of the function: $$\int \frac{dx}{1+\sin x+\cos x}$$ Answer: $$\int \frac{dx}{1+\sin x+\cos x} = \ln |2\tan(x/2)+2|+C$$

Step 1: Replace trigonometric functions and differential with given substitution relations

First, we will use the given relations A, B, and C to replace \(dx\), \(\sin x\), and \(\cos x\) in the given integral: \(\int \frac{dx}{1+\sin x+\cos x}\). According to relations A, B, and C: $$dx=\frac{2}{1+u^2}du$$ $$\sin x=\frac{2u}{1+u^2}$$ $$\cos x=\frac{1-u^2}{1+u^2}$$ Now, substitute these expressions into the integral: $$\int \frac{1}{1+\frac{2u}{1+u^2}+\frac{1-u^2}{1+u^2}} \cdot \frac{2}{1+u^2} du$$

Step 2: Simplify the integrand

To simplify the integrand, we will first combine the terms in the numerator and denominator of the fraction inside the integral, and then multiply by the \(2/(1+u^2)\) term: $$\int \frac{1}{\frac{1+u^2+2u+1-u^2}{1+u^2}} \cdot \frac{2}{1+u^2} du = \int \frac{1}{\frac{2u+2}{1+u^2}} \cdot \frac{2}{1+u^2} du$$ Now, cancel out the common \((1+u^2)\) terms in the integrand: $$\int \frac{2}{2u+2} du$$

Step 3: Integrate with respect to u

Now, we will perform the integration with respect to \(u\): $$\int \frac{2}{2u+2} du = \ln |2u+2|+C$$ where \(C\) is the constant of integration.

Step 4: Replace u with the original variable x

Finally, we will replace \(u\) with the original variable \(x\) using the substitution \(u = \tan(x/2)\): $$\ln |2\tan(x/2)+2|+C$$ So, the integral of the given function is: $$\int \frac{dx}{1+\sin x+\cos x} = \ln |2\tan(x/2)+2|+C$$

Key Concepts

Integration Techniques

Integration is a crucial concept in calculus that allows us to find areas under curves, among other things. When faced with complex integrals, different techniques can simplify the process. Two of the most common integration techniques are:

• Substitution Method: This is a powerful technique where one substitutes part of an integral with a new variable, simplifying the integral into a more standard form. Substitution is particularly useful when integrands can be simplified by recognizing a part of the expression as a derivative of another function.


• Partial Fraction Decomposition: This technique is helpful when an integrand is a rational function. By breaking down a complex fraction into simpler parts, integration can be performed on each part individually and more easily.

To solve integrals involving trigonometric functions, as shown in the original exercise, we often need to employ a specific substitution to convert the trigonometric integrand into a rational form. This can involve trigonometric identities and specific substitution formulas that transform any resulting complex trigonometric expressions into manageable algebraic expressions.

Rational Integrands

Rational integrands are functions expressed as the ratio of two polynomials. These often appear in integrals that contain trigonometric functions. By converting these functions into rational integrands, we can simplify the integration process considerably.In the given exercise, the integrand originally involved trigonometric terms: \( \frac{1}{1+\sin x+\cos x} \).
Using the substitutions provided:

• \( \sin x=\frac{2u}{1+u^2} \)
• \( \cos x=\frac{1-u^2}{1+u^2} \)

We convert these functions into rational expressions in terms of \( u \).This transformation aids in simplifying the integral to a form that is more straightforward to evaluate, like \( \frac{2}{2u+2} \), allowing for easier integration.

Change of Variables

The change of variables technique involves substituting a new variable to simplify an integral. This method is particularly useful when dealing with complex or composite transformations.In trigonometric substitution, as in the original problem, the idea is to find a substitution that replaces difficult trigonometric functions with simpler algebraic ones. Here, the substitution \( u = \tan(x/2) \) is used. This substitution transforms trigonometric functions and differentials into rational expressions:

• \( dx = \frac{2}{1+u^2} du \)
• \( \sin x = \frac{2u}{1+u^2} \)
• \( \cos x = \frac{1-u^2}{1+u^2} \)

These transformations simplify the entire integration process by converting the integrand into a form that is easier to handle. After integrating with respect to \( u \), the final step involves substituting back \( x \) using \( u = \tan(x/2) \), to present the solution in terms of the original variable. This change of variables streamlines the problem-solving process, allowing us to tackle seemingly complex integrals.