Problem 38 - Calculus Single Variable

Question

Use trigonometric identities to compute the indefinite integrals. $$ \int \sqrt{1+\cos (2 x)} d x $$

Step-by-Step Solution

Answer

\( \sqrt{2} \sin(x) + C \)

Step 1: Simplify the Trigonometric Expression

The first step is to use the trigonometric identity for \( \cos(2x) \). Recall that \( \cos(2x) = 2\cos^2(x) - 1 \). Therefore, the expression \( 1 + \cos(2x) \) can be simplified as follows:\[ 1 + \cos(2x) = 2\cos^2(x). \] Thus, the integral becomes:\[ \int \sqrt{2\cos^2(x)} \, dx. \]

Step 2: Simplify the Integrand Further

Next, realize that \( \sqrt{2\cos^2(x)} = \sqrt{2} \cdot |\cos(x)| \). For simplicity in integration, assume \( \cos(x) \geq 0 \) (considering definite intervals would specify this assumption).Thus, the integral becomes:\[ \sqrt{2} \int \cos(x) \, dx. \]

Step 3: Compute the Indefinite Integral

Now integrate the expression \( \sqrt{2} \int \cos(x) \, dx \). This integral is straightforward:\[\int \cos(x) \, dx = \sin(x) + C_1.\]Thus, the result is:\[ \sqrt{2} \cdot \sin(x) + C,\]where \( C \) represents the constant of integration.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential relationships that involve angles and functions. They serve as foundational tools in mathematics, particularly when simplifying and solving complex expressions. One such identity useful in this exercise is the double-angle identity for cosine:

• \( \cos(2x) = 2\cos^2(x) - 1 \)

This identity helps transform the integral \( \int \sqrt{1 + \cos(2x)} \, dx \) into a more manageable form. By rewriting \( 1 + \cos(2x) \) using this identity, you achieve a clearer path toward integration. Why do we use these identities? Because they replace complex expressions with simpler equivalents, facilitating easier integrations or transformations in various problems. Knowing these identities by heart greatly aids in solving trigonometric integrals efficiently.

Indefinite Integrals

Indefinite integrals represent a family of functions obtained from the anti-differentiation of a function. Unlike definite integrals, which calculate a specific numerical value over an interval, indefinite integrals include a constant of integration \( C \). This constant exists because integration is the reverse process of differentiation, where constants vanish when taking derivatives.In the given exercise, once the trigonometric expression is simplified using identities, you proceed to integrate the function. The task involves essential integration of \( \cos(x) \), known for its straight-forward outcome: \( \int \cos(x) \, dx = \sin(x) + C \). Remember, always include the constant of integration unless boundary conditions are set, which would instead properly define a definite integral outcome.

Integration Techniques

Integration techniques provide the methods needed to address various challenges posed by integrals. In this exercise, the primary technique used is simplifying the integrand through trigonometric identities. When faced with \( \sqrt{1 + \cos(2x)} \), simplifying it using the identity gives \( 2\cos^2(x) \). Simplification is crucial as complex integrands may make direct integration cumbersome or impossible.To further simplify \( \sqrt{2\cos^2(x)} \), assume certain conditions like \( \cos(x) \geq 0 \), which allows you to manage the absolute value of cosine straightforwardly as \( \cos(x) \). Once simplified, you apply the direct integration technique to \( \cos(x) \), leading to an expression that can be easily integrated step by step.The essence of integration techniques lies in strategically manipulating functions into simpler forms, either through substitutions, transformations, or identities, to facilitate a more successful integration process.