Problem 53 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{\pi / 4}(\cos 2 x+\sin 2 x) d x\)

Step-by-Step Solution

Answer

The definite integral evaluates to 1.

Step 1: Identify Substitution

To use the substitution method, let's identify what can be substituted. Looking at \(\cos 2x + \sin 2x\), we can choose a substitution that simplifies the integration.

Step 2: Choose a Substitution

Choose \( u = 2x \). This implies \( du = 2dx \) or \( dx = \frac{1}{2}du \). Adjusting the limits of integration accordingly: when \( x = 0 \), \( u = 0 \), and when \( x = \frac{\pi}{4} \), \( u = \frac{\pi}{2} \).

Step 3: Rewrite the Integral

Substitute \( u = 2x \) and \( dx = \frac{1}{2}du \) into the integral. The integral becomes:\[ \int_{0}^{\pi/2} \left( \cos u + \sin u \right) \frac{1}{2} du = \frac{1}{2} \int_{0}^{\pi/2} (\cos u + \sin u) du \]

Step 4: Evaluate the Integral

Now, integrate:\( \int(\cos u + \sin u) du \) separately. The integral of \( \cos u \) is \( \sin u \) and the integral of \( \sin u \) is \(-\cos u \). Thus,\[ \int (\cos u + \sin u) du = \sin u - \cos u + C \]

Step 5: Calculate Definite Integral

Now, evaluate the definite integral:\[ \frac{1}{2} \left[\sin u - \cos u \right]_{0}^{\pi/2} = \frac{1}{2} \left[(\sin(\pi/2) - \cos(\pi/2)) - (\sin(0) - \cos(0))\right] \]

Step 6: Simplify the Result

Simplify:\[ = \frac{1}{2} [(1 - 0) - (0 - 1)] \]\[ = \frac{1}{2} [1 + 1] \]\[ = \frac{1}{2} \times 2 = 1 \]

Step 7: State the Final Result

Therefore, the value of the definite integral \( \int_{0}^{\pi / 4} (\cos 2x + \sin 2x) dx \) is 1.

Key Concepts

Substitution Rule

The Substitution Rule is a powerful technique for evaluating complex definite integrals. It involves simplifying the integral by making a substitution that transforms it into an easier form to integrate. Here's how it works:

• Identify a part of the integrand that can be substituted with a single variable, in this case, the expression inside the integral.
• Assign this expression to a new variable, say, \( u \). For example, choose \( u = 2x \), as seen in the exercise.
• Also determine the differential \( du \). Since \( u = 2x \), then \( du = 2dx \), or \( dx = \frac{1}{2}du \).

It's important to adjust the limits of integration to match the substitution. When you substitute \( u \) for \( 2x \), also change the limits to reflect \( u \) values. This makes the problem easier to solve and beautifully links different parts of integral calculus.

Trigonometric Integration

Trigonometric Integration refers to techniques of integrating functions that involve trigonometric functions like sine and cosine. These techniques are essential because trig functions appear frequently in calculus problems.
In our exercise, the functions involved are \( \cos 2x \) and \( \sin 2x \). Through substitution, the integral \( \int(\cos u + \sin u) du \) emerges, and this can be solved using basic integration rules for trigonometric functions.

• The integral of \( \cos u \) is \( \sin u \).
• The integral of \( \sin u \) is \( -\cos u \).

By applying these simple rules, we obtain \( \int (\cos u + \sin u) du = \sin u - \cos u + C \). Trigonometric Integration is powerful and extremely useful when working with periodic functions, making calculus problems more manageable.

Calculus Techniques

Calculus Techniques encompass a wide range of methods for solving mathematical problems involving differentiation and integration. These techniques form the foundation for understanding complex systems and changes within them.
In the problem presented, several calculus techniques play a role:

• **Definite Integration**: This calculates the accumulated value of a function over a certain interval, providing a precise area under the curve. It is defined with limits of integration.
• **Simplification via Substitution**: Substitution transforms a challenging integral into a simpler form, making evaluation straightforward.

By combining these techniques, the integral \( \int_{0}^{\pi / 4}(\cos 2x + \sin 2x) dx \) simplifies significantly to \( 1 \). These methods are not just confined to textbooks—understanding them opens up powerful methods for solving a wide variety of real-world problems.