Question

For the following exercises, find the definite or indefinite integral. $$ \int_{0}^{\pi / 4} \tan x d x $$

Short answer

The definite integral is \( \frac{1}{2}\ln(2) \).

Step-by-step solution

Identify the Integral Type

The problem asks for the definite integral of \( \tan x \) from \( x = 0 \) to \( x = \frac{\pi}{4} \). This means we need to evaluate the expression \( \int_{0}^{\pi / 4} \tan x \, dx \).

Rewrite Tan Using Trigonometric Identity

The function \( \tan x \) can be rewritten using the identity \( \tan x = \frac{\sin x}{\cos x} \). This will help in finding the integral.

Use a Suitable Substitution

To find the integral of \( \frac{\sin x}{\cos x} \), we'll use the substitution method. Let \( u = \cos x \), then \( du = -\sin x \, dx \) or \( -du = \sin x \, dx \).

Change Limits of Integration

When \( x = 0 \), \( u = \cos(0) = 1 \), and when \( x = \frac{\pi}{4} \), \( u = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \). These are the new limits for the integration in terms of \( u \).

Integrate with Respect to New Variable

The integral becomes \( -\int_{1}^{1/\sqrt{2}} \frac{1}{u} \, du \), which equals \( - \left[ \ln|u| \right]_{1}^{1/\sqrt{2}} \).

Evaluate the Integral

Compute \( - \left( \ln\left|\frac{1}{\sqrt{2}}\right| - \ln|1| \right) = - \left( -\ln(\sqrt{2}) - 0 \right) = \ln(\sqrt{2}) \).

Simplify the Result

Since \( \ln(\sqrt{2}) = \frac{1}{2}\ln(2) \), the final answer is \( \frac{1}{2}\ln(2) \).

Key concepts

Trigonometric Substitution

Trigonometric substitution is a powerful technique used in calculus to simplify integrals. It involves substituting a trigonometric function for a variable to transform a difficult integral into a more manageable one.

• We often use trigonometric substitution when dealing with expressions involving square roots or trigonometric identities.
• In this problem, we use the identity \( \tan x = \frac{\sin x}{\cos x} \) to rewrite the integral \( \int \tan x \, dx \).
• This identity allows us to change the integral into a form where substitution can more easily be applied.

By choosing \( u = \cos x \), the integral becomes simpler to solve using trigonometric substitution. This choice transforms the sine function in the numerator to the differential \( du \), further making calculations easier. As we change variables, we must also change the limits of integration according to the new variable, ensuring continuity in the problem-solving process. Understanding how to choose appropriate substitutions is key in mastering this technique.

Integration Techniques

Integration techniques are strategies and methods used to find integrals, especially when they cannot be directly solved through basic antiderivatives. Some common techniques include substitution (as used in this problem), integration by parts, and partial fractions.

• Substitution involves replacing a function within the integral to make it easier to solve. This often involves recognizing patterns and using identities.
• In this exercise, we rewrite \( \tan x \) using the identity \( \frac{\sin x}{\cos x} \), which facilitates the substitution method.
• After substitution, the problem transforms into a basic integral \( -\int \frac{1}{u} \, du \), which can be solved using the natural logarithm function.

Choosing the right integration technique can make solving calculus problems more efficient. Mastery of these techniques allows for a deeper understanding of how different functions interact within an integral.

Calculus Problem Solving

Solving calculus problems involves several steps, from identifying what type of problem you have to applying the correct mathematical method to solve it. Key to problem solving in calculus is breaking down the problem into manageable steps.

• Start by analyzing the integral to understand its components. Identify if it’s a definite or indefinite integral. For this exercise, it is a definite integral, as specified by the limits \( 0 \) to \( \frac{\pi}{4} \).
• Apply known mathematical identities and choose suitable substitution to transform the integral into a simpler form for calculation.
• Once the integral is evaluated, ensure you change the limits of integration according to any substitution made.
• Cross-check each step to ensure accuracy and simplify the final result to a presentable answer.

Each calculus problem presents a unique challenge, encouraging a systematic approach to problem-solving that emphasizes logical reasoning and mathematical proficiency. By practicing these techniques consistently, students become adept at handling various integral types and are better prepared for more complex calculus challenges.