Problem 61 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{-5}^{5} x \sinh x^{2} d x\)

Step-by-Step Solution

Answer

The integral evaluates to 0.

Step 1: Identify the substitution

Let's identify a suitable substitution. Notice that the integrand contains \( \sinh(x^2) \) and its derivative involves \( x \). A good substitution is \( u = x^2 \), which means \( du = 2x \ dx \) or \( x \ dx = \frac{1}{2} du \).

Step 2: Apply the substitution

Substitute \( u = x^2 \) into the integral. The limits of integration will change as well: when \( x = -5, u = (-5)^2 = 25 \) and when \( x = 5, u = 5^2 = 25 \). Thus, the definite integral becomes \( \int_{25}^{25} \frac{1}{2} \sinh(u) \ du \).

Step 3: Evaluate the new integral

The new integral \( \int_{25}^{25} \frac{1}{2} \sinh(u) \ du \) has the same upper and lower limits (both 25). Integrating over a zero-length interval gives \( 0 \), regardless of the integrand.

Key Concepts

Definite Integrals

Definite integrals give us a number that represents the net area between a function and the x-axis over a specific interval. This concept is a cornerstone of calculus and provides insights into various applications such as physics and engineering.
In definite integrals, bounds are defined, as seen in the integral \(\int_{-5}^{5} x \sinh x^{2} d x\). Here, \(-5\) and \(5\) are the limits of integration. These limits specify the interval over which the area calculation occurs.
The integral bounds change when substitution is used, revealing the significant role they play in determining the definite value of an integral. No matter the complexity of the function, if the limits are the same, like from \(25\) to \(25\) in our example, the result is always zero. This is because there is no width to calculate over, demonstrating a unique property of definite integrals.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions and play a crucial role in many fields such as calculus, physics, and engineering. They are defined using exponential functions.
One important hyperbolic function is the hyperbolic sine, \(\sinh(x)\), which is defined as \(\sinh(x) = \frac{e^x - e^{-x}}{2}\). This function, similar to other hyperbolic functions, is used in various applications, including calculations involving catenary curves, which describe the shape of a hanging chain.
In our exercise, the integrand includes \(\sinh(x^2)\). Substituting \(u = x^2\) simplifies this expression and transforms it in a way that is easier to integrate when using properties of definite integrals.

Integral Calculus

Integral calculus focuses on the concept of integration, which is essentially the reverse process of differentiation. The integration process is vital for finding areas under curves, calculating volumes, and understanding accumulation in a given context.
The substitution method in integral calculus is a strategic approach to solve integrals that are otherwise challenging to evaluate directly. It involves changing variables to simplify the integral expression and make it more manageable.

• The substitution \(u = x^2\) in our example shows how transforming variables helps to convert complex expressions to simpler ones.
• This technique not only simplifies calculations but also encourages understanding of the relationship between different functions and their derivatives through integration.
• The careful transformation of integration limits is crucial, seen as \(-5\) to \(5\) changing to \(25\) to \(25\), reinforcing the importance of proper substitution in solving definite integrals.

Integral calculus, through methods like substitution, provides tools to efficiently evaluate complex integrals, deepening our overall understanding of calculus operations.