Problem 77 - Calculus Single Variable

Question

Evaluate the given integral by applying a substitution to a formula from a table of integrals. $$ \int \frac{\tan (t)}{\sqrt{4+\cos (t)}} d t $$

Step-by-Step Solution

Answer

The integral evaluates to \(-2 \sqrt{4 + \cos(t)} + C\).

Step 1: Identify the Appropriate Substitution

Notice that the integral involves a composition of functions, namely the tangent and cosine of \( t \). The expression under the square root, \( 4 + \cos(t) \), suggests using a substitution that simplifies this. Let \( u = \cos(t) \). Then, we have \( du = -\sin(t) \, dt \). We also observe that \( \tan(t) = \frac{\sin(t)}{\cos(t)} \).

Step 2: Express the Integral in Terms of \( u \)

Given \( u = \cos(t) \), we rewrite \( \tan(t) = \frac{\sin(t)}{u} \). The differential becomes \( dt = \frac{du}{-\sin(t)} \). Substitute these into the integral to get: \( \int \frac{\tan(t)}{\sqrt{4+\cos(t)}} \, dt = \int \frac{\frac{\sin(t)}{u}}{\sqrt{4+u}} \cdot \frac{du}{-\sin(t)} \).

Step 3: Simplify and Set Up the New Integral

The \( \sin(t) \) terms cancel out, leaving: \( -\int \frac{1}{u \sqrt{4+u}} \, du \). This is a simpler integral that we can find in a table of integrals or solve separately.

Step 4: Solve the Integral Using a Formula from the Table

Consulting a table of integrals, the integral \( \int \frac{1}{u \sqrt{a+bu}} \, du \) is solvable as a standard form integral. By comparing, we let \( a = 4 \) and \( b = 1 \), leading to the integral solution of \( -2 \sqrt{4+u} + C \) where \( C \) is a constant of integration.

Step 5: Back-Substitute \( u = \cos(t) \)

Substitute \( u = \cos(t) \) back into the expression to rewrite the result in terms of \( t \). The result becomes: \( -2 \sqrt{4+\cos(t)} + C \).

Key Concepts

Integration Techniques

Integration is a powerful mathematical method used to find areas under curves, among other applications. In this context, techniques such as substitution can simplify the integration process. To tackle integrals that involve complex compositions of functions, like those that include trigonometric identities, substitution is frequently used.

• Substitution: A method where one changes variables to simplify the integration process. By converting the originally complicated integrand into a simpler form, integration becomes more manageable.

The integral we're solving involves several layers of functions, which can be daunting at first. Substitution helps by peeling away these layers. Remember, choosing the right substitution is key. A poor choice can complicate the integral further, while a good one simplifies it.

Trigonometric Substitution

Trigonometric substitution is a specific type of substitution used in calculus to simplify integrals involving trigonometric functions. This technique leverages trigonometric identities to transform integrals into more manageable forms. Let's look at how it helps simplify our integral with functions like tangent and cosine.

• Choosing Substitution: For integrals like ours, involving terms like \( \sqrt{4+\cos(t)} \), letting \( u = \cos(t) \) simplifies the expression under the square root.
• Converting Tangent: Expressing \( \tan(t) \) in terms of \( u \) using \( \tan(t) = \frac{\sin(t)}{\cos(t)} \), aids in transforming the integral into terms of \( u \).

Trigonometric substitution, often guided by familiar trigonometric identities, transforms the problem. These identities help convert complex trigonometric integrals into algebraic forms, making them easier to solve.

Definite and Indefinite Integrals

Integrals can be classified into definite and indefinite types, each serving a unique purpose in calculus. Understanding the difference is crucial for correctly evaluating and interpreting integral solutions.

• Indefinite Integrals: Represent the family of all antiderivatives of a function. The solution includes a constant \( C \), reflecting all possible antiderivatives.
• Definite Integrals: Evaluate to a specific number and measure the net area under a curve from one point to another along the x-axis. Here, we've focused on an indefinite integral, which generally does not include specific limits of integration.

In our example, because the integral is indefinite, its solution is expressed with a constant of integration \( C \). Indefinite integrals are central in finding general solutions to antiderivatives.

Table of Integrals

A table of integrals is a valuable resource, often used to quickly find antiderivatives without performing integration from scratch. Tables list formulas for the integrals of various functions, serving as a shortcut for many problems.

• Locating Formulas: Identifying a formula in the table that matches the transformed integral helps in applying the solution directly. For example, the integral \( \int \frac{1}{u \sqrt{a+bu}} \; du \) can be found in many integral tables as a standard form.
• Applying Solutions: Using these formulas, the integration process becomes straightforward. By comparing and substituting given values for \( a \) and \( b \), you can directly obtain an antiderivative in terms of \( u \).

Using a table of integrals enhances efficiency, making it easier to solve otherwise time-consuming integrals and ensuring precision in the solutions obtained.