Problem 44 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(41-50\) . $$ \int 4 \cosh (3 x-\ln 2) d x $$

Step-by-Step Solution

Answer

\(\frac{4}{3}\sinh(3x - \ln 2) + C\)

Step 1: Recognize the integration formula

We need to integrate the function \(4\cosh(3x - \ln 2)\). Recall that the integral of \(\cosh(kx + c)\) with respect to \(x\) is \((1/k)\sinh(kx + c)\).

Step 2: Identify constants

Identify the constant \(k\) from \(\cosh(kx + c)\). Here, \(k = 3\), and the constant inside is \(-\ln 2\). We also have an outer constant 4 that we factor out of the integral.

Step 3: Apply the integration formula

We apply the integration formula to find \(\int \cosh(3x - \ln 2) \, dx\). By the formula, we get \((1/3)\sinh(3x - \ln 2) + C\).

Step 4: Multiply by the outer constant

Multiply the result of the integration by 4, which we factored out earlier. Thus, our result becomes \((4/3)\sinh(3x - \ln 2) + C\).

Step 5: Write the final answer

Combine all the steps to write the final answer as \(\int 4 \cosh(3x - \ln 2) \, dx = \frac{4}{3}\sinh(3x - \ln 2) + C\).

Key Concepts

Integration

Integration is a fundamental concept in calculus that deals with finding the area under a curve or the accumulation of quantities. It's essentially the reverse process of differentiation. When you're integrating a function, you're looking for its antiderivative. This involves identifying a function that, when differentiated, returns the original function.

One of the most important tools in integration is the integral formula. To successfully integrate, it's crucial to recognize the format of the function you're dealing with. For example, with hyperbolic functions, knowing the basic integral formulas like \[ \int \cosh(kx + c) \, dx = \frac{1}{k}\sinh(kx + c) + C \]helps in directly evaluating integrals involving hyperbolic terms.

• You identify constants and variables in the function.
• Factor out constants from the integral for simplicity.
• Apply standard formulas or integration by parts if needed.

Overall, integration is about recognizing patterns and using known formulas to solve for the antiderivative.

Hyperbolic Functions

Hyperbolic functions, like their trigonometric counterparts, are essential in math. They involve hyperbolas just as sine and cosine involve circles. The main hyperbolic functions include sinh (hyperbolic sine) and cosh (hyperbolic cosine).

The functions are defined as follows:

• \( \sinh(x) = \frac{e^x - e^{-x}}{2} \)
• \( \cosh(x) = \frac{e^x + e^{-x}}{2} \)

They're not just theoretical. Hyperbolic functions are used in real-world applications like physics and engineering, particularly in calculating forces and in hyperbolic geometry.

• They satisfy identities similar to trigonometric functions, like \( \cosh^2(x) - \sinh^2(x) = 1 \).
• They have similar derivatives and integrals as their trigonometric counterparts.

Understanding these functions is crucial when integrating, as seen in the exercise where the integral of \( \cosh \) was found using known hyperbolic identities and formulas.

Definite Integrals

Definite integrals calculate the area under a curve between two points. Unlike indefinite integrals, which produce a general antiderivative plus a constant, definite integrals result in a specific number.

The notation \( \int_{a}^{b} f(x) \, dx \) is used, where \(a\) and \(b\) are the bounds of integration. These bounds specify the interval over which the area under the curve \(f(x)\) is calculated.

• The Fundamental Theorem of Calculus connects definite integrals and derivatives.
• It states that if \(F(x)\) is an antiderivative of \(f(x)\), then \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]

This theorem simplifies finding the area by allowing us to find the difference between the antiderivative evaluated at the upper and lower bounds.
When working with definite integrals, it’s essential to apply boundaries correctly to get accurate results, making them not just a symbolic notation but a practical computation tool in calculus.