Question

Evaluate each of the following integrals by u-substitution. \(\int \tan \left(\frac{x}{2}\right) \sec ^{2}\left(\frac{x}{2}\right) d x\)

Short answer

\( \tan^2 \left(\frac{x}{2}\right) + C \)

Step-by-step solution

Identify the substitution

To solve the given integral by substitution, first identify a part of the integral that can be set as the substitution variable, \( u \). Here, set \( u = \frac{x}{2} \).

Differentiate to find du

Differentiate the substitution equation with respect to \( x \). If \( u = \frac{x}{2} \), then \( \frac{du}{dx} = \frac{1}{2} \). Thus, \( du = \frac{1}{2} dx \) or equivalently, \( dx = 2du \).

Rewrite the integral in terms of u

Substitute \( \frac{x}{2} \) with \( u \) and \( dx \) with \( 2du \) into the integral. The integral becomes: \[ \int \tan(u) \sec^2(u) \cdot 2 \, du = 2 \int \tan(u) \sec^2(u) \, du \].

Recognize the composite function structure

Notice that \( \sec^2(u) \) is the derivative of \( \tan(u) \). Thus, the integral has the form \( \int f'(u) f(u) \, du \), which simplifies to \( \frac{1}{2} f(u)^2 + C \).

Integrate using the simple formula

The integral \( 2 \int \tan(u) \sec^2(u) \ du \) simplifies to \( \tan^2(u) + C \) based on the derivative structure identified in Step 4.

Substitute back to original variable x

Replace \( u \) back with \( \frac{x}{2} \). The solution of the integral is \( \tan^2 \left(\frac{x}{2}\right) + C \).

Key concepts

u-substitution

U-substitution is a fundamental technique in calculus for simplifying the integration of complex functions. The main idea is to transform a given integral into an easier form by substituting a part of the integral with a new variable, commonly denoted as \( u \). This technique is particularly useful when dealing with functions where one part is the derivative of another.

Key Steps to U-Substitution:

• Identify a part of the integral to substitute as \( u \). Often, it's advantageous to choose a function whose derivative also appears in the integral.
• Differentiate the chosen \( u \) function to find \( du \). This involves determining the derivative \( \frac{du}{dx} \) and solving for \( du \) in terms of \( dx \).
• Rewrite the original integral in terms of \( u \) by substituting both the \( u \) variable and the \( du \) expression.
• Perform the integration in terms of \( u \).
• Substitute back the original variable into the solution to express the final result in terms of \( x \).

U-substitution simplifies the integration process, making it easier to solve more complex problems.

trigonometric functions

Trigonometric functions play a crucial role in calculus and integration, where they often appear in both simple and compound forms. Functions like sine, cosine, and tangent, along with their derivatives and integrals, are pivotal when solving calculus problems. In our problem:

Understanding Specific Functions:

• Tangent: Represented as \( \tan(u) \), this function is the ratio of the sine and cosine functions, \( \tan(u) = \frac{\sin(u)}{\cos(u)} \). Its derivative is \( \sec^2(u) \).
• Secant: The secant function \( \sec(u) \) is the reciprocal of the cosine function, \( \sec(u) = \frac{1}{\cos(u)} \). Its derivative, \( \sec(u)\tan(u) \), plays a key role in integration processes.

When handling trigonometric functions in integrals, identifying derivatives of basic identities can clarify integration processes. Recognizing such patterns, as seen in this problem, simplifies evaluation.

calculus problem

Addressing calculus problems often requires a multi-step process to achieve a solution, especially when integrals are involved. Calculus problems test the understanding of derivatives, functions, and their antiderivatives. The given problem is a perfect example of a calculus problem involving integration.

Steps in Solving Calculus Problems like Integrals:

• Analyze the Problem: Examine the given integral to understand the function components and potential relationships.
• Choose a Method: Decide on the most effective technique, like u-substitution, to break down the integral efficiently.
• Solve Step-by-Step: Follow a systematic approach to rewrite and integrate the function. This involves identifying derivative patterns and using rules of integration.
• Verify and Simplify: After finding the integral, check for consistency and simplify, substituting back any prior changes.

Handling calculus problems involves both analytic skills and the proper application of mathematical techniques to explore beyond basic arithmetic limits.