Problem 48 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int \operatorname{csch}^{2}(5-x) d x$$

Step-by-Step Solution

Answer

The integral is \(\coth(5-x) + C\).

Step 1: Recall the Relevant Integral Formula

To solve this integral, we note that the integral of \(\operatorname{csch}^2(x)\) is \(-\coth(x) + C\), where \(C\) is the constant of integration.

Step 2: Identify the Substitution

Here, we recognize that the argument of the hyperbolic function, \(5-x\), can be adjusted using substitution. Let \(u = 5 - x\).

Step 3: Differentiate the Substitution

Calculate \(du\) by differentiating the substitution: \(du = -dx\). This implies \(dx = -du\).

Step 4: Substitute and Simplify the Integral

Rewrite the integral using substitution. The integral becomes \(- \int \operatorname{csch}^2(u) du\).

Step 5: Evaluate the Integral

Use the integral formula from Step 1 to solve \(- \int \operatorname{csch}^2(u) du = \coth(u) + C\).

Step 6: Substitute Back to Original Variable

Now, replace \(u\) with \(5-x\). We get \(\coth(5-x) + C\).

Key Concepts

Hyperbolic Functions

Hyperbolic functions share similarities with the well-known trigonometric functions but are based on hyperbolas instead of circles. The key hyperbolic functions are the hyperbolic sine \( \sinh(x) \,\), cosine \( \cosh(x) \,\), and tangent \( \tanh(x) \,\). More complex functions like the hyperbolic cosecant \( \operatorname{csch}(x) \,\), depicted in our problem, are defined as reciprocals. Specifically, the hyperbolic cosecant is the reciprocal of the hyperbolic sine: \( \operatorname{csch}(x) = \frac{1}{\sinh(x)} \,\).
Hyperbolic functions often emerge in calculus, especially within integrals and differential equations. They offer unique properties, such as identities resembling those of trigonometric functions, which make them essential tools in advanced mathematics.
These functions model natural phenomena like the shape of a hanging cable, known as the catenary curve, and they describe relativistic velocities in physics. Understanding their properties enhances comprehension of many mathematical and physical processes.

Substitution Method

The substitution method is a common integration technique used to simplify the process of finding integrals. It involves replacing a part of the integral with another variable, making the integral easier to solve. In our exercise, we let \( u = 5 - x \,\) to perform the substitution, which adjusts the argument of the hyperbolic function. This transformation often simplifies the integral into a more recognizable form, such as a standard integral that we can solve directly.

• This technique is akin to reversing the chain rule applied in differentiation. It essentially rewrites the integral in terms of a new variable.
• When substituting, always remember to adjust the differential accordingly. Here, differentiating \( u = 5 - x \,\) gives \( du = -dx \,\), which means \( dx = -du \,\).
• Correct substitution and simplification pave the way to correctly solving the integral and integrating more complex expressions.

Using substitution can greatly simplify seemingly intractable integrals, making them accessible and solvable.

Integration Techniques

Mastering integration techniques is crucial in calculus to solve a wide variety of integral problems. These techniques help transform difficult integrals into simpler forms. In our case, we integrated \( \operatorname{csch}^{2}(x) \,\) by recognizing it is related to a known formula: \( \int \operatorname{csch}^2(x) \, dx = -\coth(x) + C \,\).
Having a strong grasp of standard integral formulas is invaluable. Many integrals reduce directly to these formulas or a slightly modified form. This means:

• Always refer back to standard integral formulas as a reference point.
• Check for opportunities to simplify the integration using techniques like substitution or partial fraction decomposition when applicable.
• Confirm that the transformed integral corresponds with a known integral pattern to ensure you are on the right path.

Applying these techniques efficiently can make integral calculus much more manageable and straightforward. Understanding when and how to use different strategies enhances problem-solving skills and analytical thinking.