Question

Evaluate each integral. $$\int_{\ln 2}^{\ln 3} \operatorname{csch} y d y$$

Short answer

Question: Evaluate the definite integral: $$\int_{\ln 2}^{\ln 3} \operatorname{csch} y d y$$ Answer: $$\int_{\ln 2}^{\ln 3} \operatorname{csch} y d y = \operatorname{coth}{(\ln 2)} - \operatorname{coth}{(\ln 3)}$$

Step-by-step solution

Rewrite the given function with sinh

The csch function is the reciprocal of the sinh function. So, we can rewrite the given integral as: $$\int_{\ln 2}^{\ln 3} \frac{1}{\sinh{y}} d y$$

Integration

Now we will integrate the given function. The integral of \(\frac{1}{\sinh{y}}\) is a well-known function, which is \(-\operatorname{coth}{y}\). So, we have: $$-\operatorname{coth}{y} \Big|_{\ln 2}^{\ln 3}$$

Evaluate the bounds of the integral

Now, we will substitute the bounds into the function to evaluate the definite integral: $$-\operatorname{coth}{(\ln 3)} - (-\operatorname{coth}{(\ln 2)})$$

Simplify the expression

Now, we can simplify the expression: $$\operatorname{coth}{(\ln 2)} - \operatorname{coth}{(\ln 3)}$$ This is the final answer for the given integral: $$\int_{\ln 2}^{\ln 3} \operatorname{csch} y d y = \operatorname{coth}{(\ln 2)} - \operatorname{coth}{(\ln 3)}$$

Key concepts

Hyperbolic Functions

Hyperbolic functions are similar to the trigonometric functions but are based on hyperbolas instead of circles.
They are often used in calculus and other branches of mathematics for functions that model hyperbolic geometries and describe natural phenomena.Some common hyperbolic functions include:

• Hyperbolic Sine (\(\sinh{x} = \frac{e^x - e^{-x}}{2}\)
• Hyperbolic Cosine (\(\cosh{x} = \frac{e^x + e^{-x}}{2}\)
• Hyperbolic Tangent (\(\tanh{x} = \frac{\sinh{x}}{\cosh{x}}\)
• Hyperbolic Cotangent (\(\coth{x} = \frac{\cosh{x}}{\sinh{x}}\)
• Hyperbolic Cosecant (\(\csch{x} = \frac{1}{\sinh{x}}\)
• Hyperbolic Secant (\(\sech{x} = \frac{1}{\cosh{x}}\)

Hyperbolic cosecant, or \(\operatorname{csch} x\), is particularly important in the discussed integral. It is the reciprocal of \(\sinh{x}\).
Like their trigonometric counterparts, they also have certain identities and properties that make them valuable in calculus applied problems.

Definite Integrals

The definite integral is a fundamental concept in calculus.
It is used to calculate the area under a curve between two specific points on the x-axis.When setting up a definite integral, such as \(\int_{a}^{b} f(x) \, dx\), the limits \(a\) and \(b\) indicate the range over which the area is being calculated.
This integral focuses not just on finding an antiderivative, but on evaluating that antiderivative at those bounds and subtracting the results.The process involves:

• Finding the antiderivative of the function \(f(x)\).
• Evaluating that antiderivative at the upper bound \(b\).
• Evaluating it again at the lower bound \(a\).
• Subtracting the latter result from the former. \(F(b) - F(a)\).

For our specific problem, we evaluate the integral \(\int_{\ln 2}^{\ln 3} \operatorname{csch} y \, dy\) by first expressing it in terms of \(\sinh\) and then integrating.
We substitute bounds like \(\ln 2\) and \(\ln 3\) into the resulting expression for a complete solution.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are inverses of the primary trigonometric functions.
In our problem, it's useful to understand these when converting hyperbolic functions into a solvable form for integration.Basic reciprocal identities include:

• Cosecant, which is the reciprocal of sine \(\operatorname{csc} x = \frac{1}{\sin{x}}\)
• Secant, the reciprocal of cosine \(\operatorname{sec} x = \frac{1}{\cos{x}}\)
• Cotangent, the reciprocal of tangent \(\operatorname{cot} x = \frac{1}{\tan{x}}\)

These help simplify complex expressions, facilitate integrals, and offer alternative pathways to arrive at solutions.
For instance, in tackling the given integral, transforming \(\operatorname{csch} y\) to \(\frac{1}{\sinh y}\) helped in applying the known antiderivative \(-\coth y\).
This approach highlights how reciprocal properties aid in transitioning from complicated forms to easier integrable entities.