Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated integrations. $$ \int \frac{1+\cos 2 x}{\sin ^{2} 2 x} d x $$

Step-by-Step Solution

Answer

The integral is \(-\frac{1}{2} \cot x + C\).

Step 1: Simplify the Integrand

Notice that the given integrand can be rewritten using trigonometric identities. Recall that \(1 + \cos 2x = 2\cos^2 x\), and \(\sin^2 2x = (2\sin x \cos x)^2 = 4\sin^2 x \cos^2 x\). This means the integrand becomes \(\frac{2\cos^2 x}{4\sin^2 x \cos^2 x} = \frac{1}{2\sin^2 x}\). So, our integral simplifies to \(\int \frac{1}{2\sin^2 x} \, dx\).

Step 2: Use a Trigonometric Identity

Recognize that \(\frac{1}{\sin^2 x}\) is equal to \(\csc^2 x\). Therefore, our integral becomes \(\frac{1}{2} \int \csc^2 x \, dx\).

Step 3: Integrate the Function

Recall that the integral of \(\csc^2 x\) is \(-\cot x\). Therefore, integrating gives \(\frac{1}{2} (-\cot x) + C\), where \(C\) is the integration constant.

Step 4: Simplify the Result

Simplify the expression to \(-\frac{1}{2} \cot x + C\). This is our final result for the integral.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations involving trigonometric functions that hold true for any angle. These identities simplify complex expressions into more manageable forms. They are essential tools in calculus, especially when dealing with integrals involving trigonometric functions. For example:

• One well-known identity is the double-angle formula: \(1 + \cos 2x = 2\cos^2 x\). This identity helps transform expressions that include trigonometric functions into forms that are easier to integrate.
• Another crucial identity is the expression for \(\sin^2 x\) and \(\cos^2 x\) in terms of squares: \(\sin^2 x = \frac{1 - \cos 2x}{2}\) and \(\cos^2 x = \frac{1 + \cos 2x}{2}\).

Using these identities effectively allows us to simplify integrands, making integration possible where it otherwise would not be.”
In our example, recognizing the identity \(1 + \cos 2x = 2\cos^2 x\) allowed us to transform the integrand from a complex fraction into \(\frac{1}{2\sin^2 x}\), paving the way for further simplification.

Integral Calculus

Integral calculus is an essential branch of calculus that deals with the concept of integration, which is the process of finding the integral of a function. Integration accumulates quantities, such as areas under curves, providing vital tools for solving physical and mathematical problems.Key components:

• The integral symbol \(\int\) denotes the process of integration.
• Definite integrals result in a numerical value representing accumulated quantity within certain boundaries.
• Indefinite integrals, by contrast, represent a family of functions and include a constant of integration \(C\).
• Integration can be thought of as the reverse process of differentiation.

When approaching an integral, it's crucial to identify simplifications, substitutions, or techniques, such as recognizing trigonometric identities, that make the integral feasible.”
In our example, we employ the notion of rewriting the integrand using trigonometric identities to transform it into a simpler form. Thus, we proceed by performing an antiderivative operation on \(\frac{1}{2\sin^2 x}\), ultimately finding the integral involving \(\csc^2 x\).

Indefinite Integrals

Indefinite integrals represent a fundamental concept in calculus, involving the integration of a function without specified limits or bounds. They yield an entire family of possible functions, representing accumulated changes or areas up to any given point.Important elements include:

• An indefinite integral is expressed as \(\int f(x) \, dx = F(x) + C\), where \(F(x)\) is the antiderivative, and \(C\) is an arbitrary constant of integration.
• The constant \(C\) accounts for any vertical shift in the family of curves, as integration is an inverse process of differentiation.
• Recognizing common derivatives can help find the correct antiderivative quickly.

Using indefinite integrals, we solve a range of problems by reversing the process of differentiation and understanding the behavior of functions over any interval. In our example, identifying the expression \(\csc^2 x\) allowed for simple integration since the antiderivative \(-\cot x\) is well-known from trigonometric derivatives."