Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-54, perform the indicated integrations. \(\int \frac{\csc ^{2} 2 t}{\sqrt{1+\cot 2 t}} d t\)

Step-by-Step Solution

Answer

The integral evaluates to \(-\sqrt{1+\cot 2t} + C\).

Step 1: Simplifying the Integrand

First, recognize that the integrand \( \frac{\csc^2 2t}{\sqrt{1+\cot 2t}} \) can be rewritten using trigonometric identities. Recall that \( \csc 2t = \frac{1}{\sin 2t} \) and \( \cot 2t = \frac{\cos 2t}{\sin 2t} \). This gives us steps to express \( \csc^2 2t \) and \( \cot 2t \) in simpler trigonometric terms.

Step 2: Substitution to Simplify the Integration

To solve the integral, consider a trigonometric substitution. Let \( u = \cot 2t \). Then, the derivative \( du = -2 \csc^2 2t \ dt \) allows us to reformulate the integral in terms of \( u \). Rewriting the differential gives \( dt = \frac{du}{-2 \csc^2 2t} \).

Step 3: Setting up the New Integral

Using the substitution \( u = \cot 2t \), the integral changes from \( \int \frac{\csc^2 2t}{\sqrt{1+\cot 2t}} dt \) to \( \int \frac{1}{\sqrt{1+u}} \left(-\frac{1}{2}\right) du \). Simplifying further, we have \(-\frac{1}{2} \int \frac{1}{\sqrt{1+u}} du\).

Step 4: Solving the Integral

Now solve the integral \( -\frac{1}{2} \int \frac{1}{\sqrt{1+u}} du \). Recognize that this is a standard integral that results in \(-\frac{1}{2} \cdot 2 \sqrt{1+u} + C\), which simplifies to \(-\sqrt{1+u} + C\).

Step 5: Back-substitution

Re-substitute \( u = \cot 2t \) back into the solution of the integral to transform it in terms of \( t \). The result becomes \(-\sqrt{1 + \cot 2t} + C\).

Step 6: Final Solution

Combine all the solutions for each step to write the final solution of the integral. The integral \( \int \frac{\csc^2 2t}{\sqrt{1+\cot 2t}} dt \) resolves to \(-\sqrt{1+\cot 2t} + C\), where \( C \) is the constant of integration.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is an important technique in calculus to simplify integrals involving square roots and trigonometric functions. This method can convert complex integrals into simpler ones that are easier to solve. By redefining variables using trigonometric identities, we can transform the structure of an integral.

In this problem, a substitution was made with \( u = \cot 2t \). This substitution was strategic because it simplified the integrand substantially. The differentiation yielded \( du = -2 \csc^2 2t \, dt \), which allowed us to express everything in terms of \( u \). This transformation translated the original complex integral into a more straightforward one that was feasible to integrate with standard techniques.

Benefits of trigonometric substitution include:

• Converting difficult integrals into easier forms.
• Making use of known derivatives and integrals.
• Simplifying computations by eliminating complex functions.

When facing integrals with square roots and trigonometric functions, this method is often one of the first tactics to attempt.

Integral Calculus

Integral calculus is a fundamental branch of calculus focusing on the concept of integration. Integration allows us to find areas under curves, among other applications. Depending on the form of an integral, various techniques like direct integration, substitution, and parts can be applied to solve it.

In our exercise, the integral \( \int \frac{\csc^2 2t}{\sqrt{1+\cot 2t}} dt \) required the use of trigonometric substitution. The transformation reduced the complexity, which made it possible to solve using known integrals. Integration in calculus often involves such transformations to utilize simpler formulae and identities to solve complex problems.

Overview of integral calculus includes:

• Finding antiderivatives or indefinite integrals.
• Evaluating definite integrals for the exact area under curves.
• Applying transformations, like substitutions, to solve complex equations.

Integral calculus is essential for understanding continuous accumulations and additions in various fields, including physics, engineering, and economics.

Trigonometric Identities

Trigonometric identities are equations that relate the angles and functions of trigonometry, such as sine, cosine, and tangent. These identities are valuable tools in simplifying expressions and solving trigonometric problems, often used in calculus to assist in integration tasks.

In this exercise, identities like \( \csc 2t = \frac{1}{\sin 2t} \) and \( \cot 2t = \frac{\cos 2t}{\sin 2t} \) were used to rewrite the integrand. These simplifications enable us to see a clearer path to perform substitutions or transformations. Such manipulations are vital for solving complicated integrals as they help reveal relationships and simplify expressions.

Common trigonometric identities include:

• Pythagorean identities like \( \sin^2 x + \cos^2 x = 1 \).
• Reciprocal identities such as \( \csc x = \frac{1}{\sin x} \).
• Angle sum and difference identities.

Knowing these identities can immensely help in both simplifying integrals and providing pathways to solution.