Evaluate the following integrals.
∫ (1 - cosx)/(1 + cosx) dx

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Step 1: Recognize that the integral involves trigonometric functions. To simplify the integrand, consider using trigonometric identities. Specifically, recall that 1 + cos(x) can be expressed as 2cos²(x/2) and 1 - cos(x) can be expressed as 2sin²(x/2). This substitution will help simplify the fraction.

Step 2: Rewrite the integrand using the trigonometric identities: ∫ (1 - cos(x))/(1 + cos(x)) dx = ∫ (2sin²(x/2))/(2cos²(x/2)) dx. Simplify the fraction to get ∫ tan²(x/2) dx.

Step 3: Use the identity tan²(u) = sec²(u) - 1 to rewrite the integrand. This gives ∫ tan²(x/2) dx = ∫ (sec²(x/2) - 1) dx.

Step 4: Split the integral into two parts: ∫ sec²(x/2) dx - ∫ 1 dx. The first term, ∫ sec²(x/2) dx, integrates to (2tan(x/2)) because the derivative of tan(x/2) is sec²(x/2) multiplied by the chain rule factor of 1/2. The second term, ∫ 1 dx, integrates to x.

Step 5: Combine the results from the two integrals and include the constant of integration, C. The final expression will be in terms of x and tan(x/2).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration

Integration is a fundamental concept in calculus that involves finding the accumulated area under a curve represented by a function. It is the reverse process of differentiation and can be used to calculate definite and indefinite integrals. Understanding integration techniques, such as substitution and integration by parts, is essential for solving integral problems.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are crucial for simplifying expressions and solving integrals involving trigonometric functions. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas, which can help in transforming the integrand into a more manageable form.

Substitution Method

The substitution method is a technique used in integration to simplify the integrand by changing variables. This method involves substituting a part of the integral with a new variable, which can make the integral easier to evaluate. It is particularly useful when dealing with composite functions or when the integrand can be expressed in a simpler form through a suitable substitution.

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Textbook Question

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.

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Textbook Question

85. Another form of ∫ sec x dx

a. Verify the identity:

sec x = cos x / (1 - sin² x)

b. Use the identity in part (a) to verify that:

∫ sec x dx = (1/2) ln |(1 + sin x)/(1 - sin x)| + C

Textbook Question

6. Evaluate ∫ cos x √(100 − sin² x) dx using tables after performing the substitution u = sin x.

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.