Problem 56 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(41-60\) $$\int_{0}^{\pi / 2} 2 \sinh (\sin \theta) \cos \theta d \theta$$

Step-by-Step Solution

Answer

The integral evaluates to \( 2 (\cosh(1) - 1) \).

Step 1: Identify the Integral and Substitution

The given integral is \( \int_{0}^{\pi / 2} 2 \sinh (\sin \theta) \cos \theta \, d\theta \). To simplify, use substitution: let \( u = \sin \theta \), which implies \( du = \cos \theta \, d\theta \). Update the limits of integration: when \( \theta = 0 \), \( u = \sin 0 = 0 \), and when \( \theta = \pi/2 \), \( u = \sin(\pi/2) = 1 \).

Step 2: Rewrite the Integral in Terms of 'u'

Substitute the expressions to change the integral's variable to \( u \). The integral becomes \( 2 \int_{0}^{1} \sinh(u) \, du \).

Step 3: Integrate with Respect to 'u'

Recall that the integral of \( \sinh(u) \) is \( \cosh(u) \). Thus, the integral becomes \( 2 [\cosh(u)]_{0}^{1} \).

Step 4: Evaluate the Definite Integral

Apply the Fundamental Theorem of Calculus to evaluate the definite integral: \( 2 [\cosh(1) - \cosh(0)] \). Since \( \cosh(0) = 1 \) and \( \cosh(1) \) is a constant, the expression simplifies to \( 2 (\cosh(1) - 1) \).

Step 5: Final Calculation and Conclusion

The value of \( 2 [\cosh(1) - 1] \) is the final result for the given integral. This is a numerical value, calculated using \( \cosh(1) \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions, much like trigonometric functions, play a crucial role in calculus. They are analogous to the sine and cosine functions but are based on hyperbolas instead of circles. The primary hyperbolic functions are the hyperbolic sine (\(\sinh(x)\)) and hyperbolic cosine (\(\cosh(x)\)). These functions are defined as:

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

Hyperbolic functions show up in many areas of mathematics, particularly in the solution of differential equations and the description of hanging cables or chains in physics, known as catenaries.
An important property of these functions is their similarity to the trigonometric identities. For instance, hyperbolic functions satisfy the identity: \(\cosh^2(x) - \sinh^2(x) = 1\).In the exercise, we encounter the term \(\sinh(\sin \theta)\), which combines both hyperbolic and trigonometric functions. Understanding their behavior and calculating their integrals is often part of more complex integration processes.

U-Substitution

U-substitution is a powerful technique used in integral calculus to simplify the process of integration. It works by substituting a part of the integral with a single variable, typically 'u', which allows us to transform the integral into a simpler form. This process is akin to the reverse of the chain rule in differentiation.To apply u-substitution, follow these steps:

• Identify a part of the integrand that can be replaced with 'u'.
• Find the derivative of 'u' with respect to the original variable, which gives us \(du\).
• Change the limits of integration according to the new variable.
• Rewrite the integral in terms of 'u' and integrate.

In our exercise, we used the substitution \(u = \sin \theta\), which simplifies integration with respect to \(\theta\). This substitution not only simplified the integral but also allowed us to directly calculate the integral of the hyperbolic function \(\sinh(u)\).
By practicing u-substitution, students can gain confidence and ability in tackling more complex integrals. It is essential to understand this technique thoroughly, as it forms a foundation for further learning in calculus.

Definite Integrals

Definite integrals are a fundamental concept in calculus, used to calculate the accumulation of quantities, such as areas under curves and total changes over an interval. Unlike indefinite integrals, which yield a general form of antiderivatives, definite integrals provide a specific numerical value.To calculate a definite integral, apply the Fundamental Theorem of Calculus, which connects differentiation and integration by stating that:

• If \(F\) is an antiderivative of \(f\) on an interval \([a, b]\), then \(\int_{a}^{b} f(x) \, dx = F(b) - F(a)\).

When dealing with hyperbolic functions, the procedure remains consistent. In our context, we calculated the definite integral of \(2 \sinh(\sin \theta) \cos \theta\). After applying u-substitution, it transformed into \(2 \int_{0}^{1} \sinh(u) \, du\), which we then evaluated using the antiderivative \(\cosh(u)\).Understanding definite integrals and their computations is vital, as they have numerous applications in physics, engineering, and economics. They help in determining exact values for quantities in real-world problems, such as work done by a force or the total income over a period.