Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. $$\int_{\ln \frac{\pi}{4}}^{\ln \frac{\pi}{2}} e^{w} \cos e^{w} d w$$

Short answer

Question: Use integration by parts to find the definite integral of the given function: $$\int_{\ln \frac{\pi}{4}}^{\ln \frac{\pi}{2}} e^{w} \cos e^{w} d w$$ Answer: The integral after applying integration by parts is: $$\int_{\ln \frac{\pi}{4}}^{\ln \frac{\pi}{2}} e^{w} \cos e^{w} d w = e^{w} \cos e^{w} + \int_{\ln \frac{\pi}{4}}^{\ln \frac{\pi}{2}} e^{2w} \sin e^{w} d w$$ The new integral on the right-hand side can be approximated using numerical methods, but cannot be solved analytically.

Step-by-step solution

Determine u and dv

Let \(u = \cos e^w\) and \(dv = e^w dw\)

Finding du

Differentiate \(u\) with respect to \(w\): $$du = -\sin e^{w} e^{w} d w$$

Finding v

Integrate \(dv\) with respect to \(w\): $$v = \int e^w dw = e^w$$

Applying Integration by Parts

Using the formula for integration by parts: $$\int u dv = uv - \int v du$$ Substitute the values of \(u, du, v\): $$\int_{\ln \frac{\pi}{4}}^{\ln \frac{\pi}{2}} e^{w} \cos e^{w} d w = e^{w} \cos e^{w} - \int_{\ln \frac{\pi}{4}}^{\ln \frac{\pi}{2}} e^w (-\sin e^{w} e^{w}) d w$$

Simplify expression

Now the expression becomes: $$\int_{\ln \frac{\pi}{4}}^{\ln \frac{\pi}{2}} e^{w} \cos e^{w} d w = e^{w} \cos e^{w} + \int_{\ln \frac{\pi}{4}}^{\ln \frac{\pi}{2}} e^{2w} \sin e^{w} d w$$ This new integral on the right-hand side is difficult to solve analytically. However, we can approximate its value using numerical methods like Simpson's rule or the trapezoidal rule. For the sake of simplicity, we won't go into the details of those methods in this answer.

Key concepts

Integration by Parts

Integration by parts is a clever technique derived from the product rule of differentiation which helps us tackle integrals of products of functions. It is based on the formula:\[\int u \, dv = uv - \int v \, du\]Here's how it works in a simple way:

• First, you choose which part of the integrand will be \(u\) and which will be \(dv\).
• Differentiate \(u\) to find \(du\), and integrate \(dv\) to find \(v\).
• Plug these into the integration by parts formula.
• The goal is that the new integral, \(\int v \, du\), is simpler to solve.

In the context of our exercise, we identified \(u = \cos e^w\) and \(dv = e^w \, dw\). By finding \(du\) and \(v\), we created a new integral that is simpler or at least more manageable. This method is a useful tool especially when dealing with complicated functions that multiply each other within integrals.

Change of Variables

The change of variables, also known as substitution, is a powerful method that simplifies the process of integration by transforming it into a more familiar or solvable form.The idea is to choose a new variable, say \(t\), related to \(w\) such that the integral becomes easier. Here's how it goes down:

• Pick a substitution \(t = \phi(w)\) that simplifies the integral.
• Express \(dw\) in terms of \(dt\), i.e., find \(d\phi(w)/dt\).
• Rewrite the limits, if necessary, using the new variable.
• Perform the integration with respect to \(t\) and change back to \(w\) if needed.

In our specific problem, even though a change of variables wasn't explicitly used, understanding this technique allows for great flexibility when evaluating complex integrals.Additionally, if integration by parts doesn't fully simplify the integral, a followed substitution might be the key to breaking down the problem further.

Numerical Integration

When integrals are too challenging to solve analytically, we turn to numerical integration. This approach provides approximate solutions, making it extremely useful for complicated functions or when exact answers aren’t necessary. Several methods exist:

• Trapezoidal Rule: This approximates the area under the curve as a series of trapezoids and sums their area.
• Simpson's Rule: A more accurate method than trapezoidal, it approximates the curve using parabolas instead of straight lines.
• Monte Carlo Integration: Best for multi-dimensional integrals; it uses random sampling to estimate the integral.

In our context, the remaining integral after applying integration by parts is not easy to handle analytically, hence suggesting a numerical method. Using tools like Simpson's Rule or Trapezoidal Rule would give us a practical estimate for solutions. Understanding these methods equips us to tackle integrals that might otherwise appear unsolvable using standard calculus techniques.