Problem 53 - Calculus

Question

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 value of the integral is 1.

Step 1: Identify the Substitution

For the integral \( \int_{0}^{\pi / 4}(\cos 2 x + \sin 2 x) \, dx \), notice that both terms involve \(2x\). This suggests that a substitution involving \(u = 2x\) could simplify the expression.

Step 2: Calculate the Differential

Using the substitution \(u = 2x\), we calculate the differential: \(du = 2 \, dx\) or equivalently, \(dx = \frac{1}{2} du\).

Step 3: Adjust the Limits of Integration

With \(u = 2x\), adjust the limits of integration. When \(x = 0\), \(u = 2 \times 0 = 0\). When \(x = \frac{\pi}{4}\), \(u = 2 \times \frac{\pi}{4} = \frac{\pi}{2}\).

Step 4: Substitute and Simplify the Integral

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

Step 5: Evaluate the Integral

Separate the integral and solve:\[ \frac{1}{2} \left( \int_{0}^{\pi/2} \cos u \, du + \int_{0}^{\pi/2} \sin u \, du \right) \].The integral of \(\cos u\) is \(\sin u\) from \(0\) to \(\frac{\pi}{2}\):\( [\sin u]_{0}^{\pi/2} = \sin \frac{\pi}{2} - \sin 0 = 1 - 0 = 1 \).The integral of \(\sin u\) is \(-\cos u\) from \(0\) to \(\frac{\pi}{2}\):\( [-\cos u]_{0}^{\pi/2} = -\cos \frac{\pi}{2} + \cos 0 = 0 + 1 = 1 \).

Step 6: Combine and Simplify

Combine the results of the two integrals: \(1 + 1 = 2\).Since we have a factor of \(\frac{1}{2}\) from our substitution:\[ \frac{1}{2} \times 2 = 1 \].

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus, used to find the area under a curve over a specific interval. Unlike indefinite integrals, which provide a general form or family of antiderivatives, definite integrals provide a numerical value representing this area. This numerical value is crucial for analyzing the total accumulation of quantities like distance and probabilities in various fields. In our example, evaluating the integral of \[\int_{0}^{\pi / 4}(\cos 2 x + \sin 2 x) \, dx\] involves definite integration over the interval from 0 to \(\frac{\pi}{4}\).

• The area between the curve and the x-axis from 0 to \(\frac{\pi}{4}\) is calculated to get a specific numerical result.
• The limits (0 to \(\frac{\pi}{4}\)) are replaced by new ones when using substitution, which simplifies the integral to one more easily solvable between new bounds, 0 to \(\frac{\pi}{2}\) in this case.

Integrating gives us a specific value of 1, representing the accumulation of all changes described by the function within the given limits.

Integration Techniques

Integration techniques are essential tools in calculus for finding antiderivatives or integrals of functions. Among these, the substitution rule is a powerful method, especially useful in dealing with complicated expressions. The substitution rule involves changing variables to simplify integration. In this exercise, the substitution \(u = 2x\) was adopted, transforming the original function into a more manageable one.

• This substitution simplifies the integral, allowing us to break it down into smaller parts that are easier to integrate.
• The integral was split as \((\int_{0}^{\pi/2} (\cos u + \sin u)\, du)\).
• We substituted back the original variable after integration if needed, as the substitution sufficiently simplified the integration over adjusted limits \(u\).

The technique is not only beneficial in solving trigonometric integrals but is also applicable to a wide range of functions requiring integration.

Trigonometric Integrals

Trigonometric integrals involve functions that include trigonometric expressions like sine and cosine. These integrals often appear in calculus problems due to the periodic nature and wide prevalence of trigonometric functions in modeling real-world scenarios. In this problem, we face an integral with both cosine and sine functions: \(\int_{0}^{\pi / 4}(\cos 2 x + \sin 2 x) \, dx\).

• To evaluate such integrals, techniques like substitution can be extremely useful. In our case, substituting \(u = 2x\) simplifies the trigonometric integral considerably.
• By transforming \(2x\) to \(u\), it becomes easier to find the antiderivatives of \(\cos u\) and \(\sin u\), cutting down the integration task to known standard forms.

Remember that understanding the standard antiderivatives of \(\cos u\) and \(\sin u\), i.e., \(\sin u\) and \(-\cos u\) respectively, is essential in solving such integrals. Trigonometric integrals often require such identity transformations for efficient evaluation.