Problem 82 - Calculus Single Variable

Question

Evaluate the given integral by applying a substitution to a formula from a table of integrals. $$ \int \cos (\ln (t)) d t $$

Step-by-Step Solution

Answer

\( \frac{t}{2}\cos(\ln(t)) + C \)

Step 1: Choose a Substitution

Notice the integral involves the composition function \( \cos(\ln(t)) \). Set \( u = \ln(t) \). Then, \( du = \frac{1}{t} dt \). Rearranging gives \( dt = t \cdot du \). Since \( t = e^u \), substitute: \( dt = e^u \cdot du \).

Step 2: Substitute into the Integral

Substitute \( u = \ln(t) \) and \( dt = e^u du \) into the integral:\[\int \cos(\ln(t)) dt = \int \cos(u) \cdot e^u du\]

Step 3: Use Table of Integrals

Look for a formula from a table of integrals that matches \( \int \cos(u) e^u du \). The formula is:\[\int e^u (A \cos(u) + B \sin(u)) du = e^u (C \cos(u) + D \sin(u)) + C\]where \( C = \frac{A}{2} \) and \( D = \frac{B}{2} \). For our problem, \( A = 1 \) and \( B = 0 \).

Step 4: Apply the Formula

Since \( A = 1 \) and \( B = 0 \), apply the formula:\[\int e^u \cos(u) du = e^u\left(\frac{1}{2}\cos(u)\right) + C\]

Step 5: Substitute Back in Terms of t

Return to the variable \( t \) using \( u = \ln(t) \) and \( e^u = t \):\[= t\left(\frac{1}{2}\cos(\ln(t))\right) + C = \frac{t}{2}\cos(\ln(t)) + C\]

Key Concepts

Indefinite Integrals

Indefinite integrals, also known as antiderivatives, are functions that represent a family of all possible integrals of a given function. When performing an indefinite integral, we do not have defined boundaries or limits, which means the result includes an arbitrary constant, usually denoted as 'C'. This constant accounts for the fact that many functions can differ by a constant and still have the same derivative.

For example, \[\int f(x) \, dx = F(x) + C\]The indefinite integral process seeks a function, like \( F(x) \), whose derivative is the integrand, \( f(x) \). Understanding indefinite integrals is crucial because it lays the groundwork for many calculus applications, such as computing areas under curves or solving differential equations.

In the context of our exercise, we find the indefinite integral of \(\cos(\ln(t))\) with respect to \(t\), resulting in:
\[\frac{t}{2}\cos(\ln(t)) + C\]
Here, \(C\) represents the constant of integration that can vary based on initial conditions or boundary values if they are provided.

Trigonometric Integrals

Trigonometric integrals involve functions of trigonometric expressions. These integrals often appear in the integration of periodic functions or when dealing with oscillatory motions.

Trigonometric integrals can include functions like \( \sin(x) \), \( \cos(x) \), \( \tan(x) \), and their inverses. A typical approach to solving these integrals is by using identities or substitutions that simplify the integrand.

• The Pythagorean identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Angle sum and difference identities
• Product-to-sum identities

In the exercise, we dealt with the integral \( \int \cos(\ln(t)) \, dt \). Through substitution, we converted it to \( \int \cos(u) \cdot e^u \, du \), which enabled us to utilize a specific trigonometric integral formulation: \[\int e^u (A \cos(u) + B \sin(u)) \, du\]This strategy rests on identifying and restructuring parts of the integrand to conform to known formulas, which simplifies the integration process.

Integration Techniques

Integration techniques are strategies or methods used to simplify and solve integrals. Different techniques are useful depending on the form and complexity of the integrand.

Some common integration techniques include:

• Substitution: Involves changing variables to simplify the integral. It's useful for functions that are compositions of other functions.
• Integration by parts: Based on the product rule of differentiation, useful when the integrand is a product of two functions.
• Partial fraction decomposition: Breaking down complex rational functions into simpler fractions.
• Trigonometric identities: Utilizing various identities to transform or simplify trigonometric integrals.

In our problem, we used the substitution method, a foundational integration technique. The integrand \( \cos(\ln(t)) \) was simplified by substituting \( u = \ln(t) \), transforming the integral into a form suitable for applying a known table of integrals. By converting the original variable to \( u \), we made the integral easier to manage and solve. This method highlights how substitution facilitates dealing with complex compositions of functions, streamlining the process of finding antiderivatives.