7–84. Evaluate the following integrals.
82. ∫ 1/(1 + tanx) dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves the function \( \frac{1}{1 + \tan(x)} \). To simplify this, recall the trigonometric identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Substitute \( \tan(x) \) into the denominator.
2. Step 2: Rewrite the denominator as \( 1 + \tan(x) = 1 + \frac{\sin(x)}{\cos(x)} \). Combine terms under a common denominator to get \( \frac{\cos(x) + \sin(x)}{\cos(x)} \). The integral now becomes \( \int \frac{\cos(x)}{\cos(x) + \sin(x)} dx \).
3. Step 3: To simplify further, let \( u = \cos(x) + \sin(x) \). Compute \( du \) by differentiating \( u \): \( du = -\sin(x) dx + \cos(x) dx \), or equivalently \( du = (\cos(x) - \sin(x)) dx \).
4. Step 4: Notice that the numerator \( \cos(x) dx \) can be expressed in terms of \( du \). Factor out \( \cos(x) \) and rewrite the integral in terms of \( u \): \( \int \frac{1}{u} du \).
5. Step 5: Solve the integral \( \int \frac{1}{u} du \), which is a standard logarithmic integral. The result is \( \ln|u| + C \), where \( u = \cos(x) + \sin(x) \). Substitute back \( u \) to express the solution in terms of \( x \).

Key Concepts

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 is used to compute definite and indefinite integrals. Understanding integration techniques, such as substitution and integration by parts, is essential for evaluating integrals like ∫ 1/(1 + tanx) dx.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are essential in calculus, particularly when dealing with integrals involving angles. The function tan(x) is defined as the ratio of the opposite side to the adjacent side in a right triangle. Recognizing the relationships between trigonometric functions and their identities can simplify the integration process, especially in integrals involving tan(x).

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. For the integral ∫ 1/(1 + tanx) dx, using a substitution related to the tangent function can help transform the integral into a more manageable form.