Question

Perform the indicated integrations. $$ \int_{-\pi / 4}^{9 \pi / 4} e^{\cos z} \sin z d z $$

Short answer

The integral is 0.

Step-by-step solution

Determine if substitution is necessary

The integral \( \int e^{\cos z} \sin z \, dz \) is a candidate for substitution because of the presence of the function \( e^{\cos z} \) and its derivative \( \sin z \). We can consider substituting \( u = \cos z \).

Perform substitution

Let \( u = \cos z \), then \( du = -\sin z \, dz \). This implies \( -du = \sin z \, dz \). The limits of integration will change as follows: for \( z = -\pi/4 \), \( u = \cos(-\pi/4) = \sqrt{2}/2 \) and for \( z = 9\pi/4 \), \( u = \cos(9\pi/4) = \sqrt{2}/2 \) since the cosine function is periodic with period \( 2\pi \).

Rewrite the integral in terms of u

The integral becomes \( \int_{\sqrt{2}/2}^{\sqrt{2}/2} e^u (-du) \). Since the limits of integration are the same, the integral evaluates to zero.

Evaluate the definite integral

Since the limits of integration are the same, the definite integral evaluates as follows: \[ \int_{\sqrt{2}/2}^{\sqrt{2}/2} e^u (-du) = 0 \]

Key concepts

Substitution in Integration

Substitution in integration is a powerful technique often used to simplify complex integrals by transforming them into a more familiar form. This is analogous to using a substitution in algebra to solve equations.
When you encounter an integral that seems complicated due to the composition of functions, substitution can be a helpful approach. In this exercise, we see an integral of the form \( \int e^{\cos z} \sin z \, dz \). Here, we identify that the exponential function \( e^{\cos z} \) and its derivative, \( \sin z \), make it a perfect candidate for substitution.
The key steps for substitution include:

• Identifying the substitution candidate: Choose a function within the integral (often inside a composite function) whose derivative is also part of the integrand. In this case, \( u = \cos z \).
• Expressing \( dz \): Here \( du = -\sin z \, dz \), giving us \( dz = -\frac{du}{\sin z} \).
• Transforming the limits of integration: Adjust the limits so that they match the new variable, resulting in limits from \( \sqrt{2}/2 \) to \( \sqrt{2}/2 \).
• Performing the integration: Transform the original integral using substitution, evaluate, then substitute back if necessary. However, in our case, it was simplified to zero since the limits were the same.

Understanding and utilizing substitution helps to transform unwieldy integrals into manageable forms, making it an indispensable tool in calculus.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. A classic example of a periodic function is the trigonometric function \( \cos(z) \), which has a period of \( 2\pi \). This means the function repeats its values every \( 2\pi \) radians.
This property of trigonometric functions is crucial when dealing with definite integrals over an interval that includes one or more full periods of the function.
In the original exercise, the bounds of the integral shift the function through more than one full period of \( \cos(z) \). Because the function values repeat, evaluations at different angles across full periods give the same results.
Important points about periodic functions include:

• The minimum interval necessary for the function to repeat is known as the period.
• This repeatability can simplify computations, especially in definite integrals.
• In any integral stretched across full periods, contributions that cancel out can be quickly identified, sometimes leading to simpler results such as zero.

Using periodic functions in integration can substantially reduce the effort needed to evaluate integrals, especially when the period aligns neatly with the integration limits.

Change of Variables

The change of variables is a crucial concept in calculus that permits the transformation of integration problems into more convenient forms. It is very related to substitution, but specifically focuses on adjusting variables and limits of integration to accommodate the new variable system.
In the given problem, changing variables with \( u = \cos z \) allows for the transformation of the integral in terms of \( u \). This change is especially useful when the original integral appears too complex or cumbersome to handle directly.
The process for changing variables involves:

• Selecting an appropriate substitution that simplifies the integral.
• Recalculating the differential, so in this case, \( du = -\sin z \, dz \).
• Adjusting the limits of integration to reflect the new variable's viewpoint (initially from \( z \) values to \( u \) values).
• Revisiting the expression in the context of its new limits, which sometimes reveals straightforward solutions.

The change of variables not only reconfigures the integral but can also illuminate underlying symmetries or periodicities that might be hidden in the original problem. As seen in our example, this method revealed that the integral simplified to zero due to identical limits, illustrating the power of a well-chosen change of variables.