Problem 52 - Thomas Calculus

Question

Evaluate the integrals. \begin{equation}\int_{0}^{\sqrt{\ln \pi}} 2 x e^{x^{2}} \cos \left(e^{x^{2}}\right) d x\end{equation}

Step-by-Step Solution

Answer

The value of the integral is \(-\sin(1)\).

Step 1: Identify the Integral Type

The integral we need to evaluate is \( \int_{0}^{\sqrt{\ln \pi}} 2x e^{x^{2}} \cos\left(e^{x^{2}}\right) dx \). This is a definite integral, and it involves a product of functions including an exponential function, a trigonometric function, and a polynomial.

Step 2: Choose an Appropriate Substitution

To simplify the integral, use substitution. Notice the presence of \(e^{x^2}\) and the factor \(2x\), which suggests the substitution \(u = e^{x^2}\). Then, \(du = 2x e^{x^2} dx\). This substitution simplifies the integral.

Step 3: Change the Limits of Integration

Calculate the new limits of integration. When \(x = 0\), \(u = e^{0^2} = e^0 = 1\). When \(x = \sqrt{\ln \pi}\), \(u = e^{(\sqrt{\ln \pi})^2} = e^{\ln \pi} = \pi\). So the limits change from \(0\) to \(\sqrt{\ln \pi}\) in terms of \(x\) to \(1\) to \(\pi\) in terms of \(u\).

Step 4: Simplify the Integral Using Substitution

Substituting into the integral, it becomes \( \int_{1}^{\pi} \cos(u) \, du \). This is a simpler integral to evaluate.

Step 5: Evaluate the Integral

The integral \( \int \cos(u) \, du \) is a standard integral and evaluates to \( \sin(u) + C \) where \(C\) is the constant of integration, which we omit due to evaluating a definite integral. Apply this from \(1\) to \(\pi\): \([\sin(u)]_{1}^{\pi} = \sin(\pi) - \sin(1) \).

Step 6: Calculate the Definite Integral

Calculate \( \sin(\pi) - \sin(1) \). Since \( \sin(\pi) = 0 \), the result is \(-\sin(1)\).

Step 7: Conclusion: State the Final Answer

Thus, the value of the original integral \( \int_{0}^{\sqrt{\ln \pi}} 2x e^{x^{2}} \cos(e^{x^{2}}) \, dx \) is \(-\sin(1)\).

Key Concepts

Trigonometric Integration

Trigonometric integration is an essential technique in calculus for solving integrals that involve trigonometric functions. When we encounter integrals with functions like \( \cos(x) \), \( \sin(x) \), or \( \tan(x) \), we use trigonometric identities to simplify the computation. These identities help us manipulate the expressions into forms that are easier to integrate.
For example, in the problem we evaluated, the integral included \( \cos(e^{x^2}) \). By recognizing the trigonometric component, we could simplify the integral significantly.
To solve such integrals, it's crucial to understand basic trigonometric identities like:

• \( \sin^2(x) + \cos^2(x) = 1 \)
• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)
• \( \cos(2x) = \cos^2(x) - \sin^2(x) \)

For definite integrals involving trigonometric functions, we determine constants by evaluating limits, helping us find precise numerical values, as achieved by evaluating \( [\sin(u)]_1^{\pi} \) in the solution.

Substitution Method

The substitution method is a powerful technique for integrals where direct integration is complicated. It involves substituting a part of the integral with a new variable, thus transforming the integral into a simpler form.
For instance, in the given exercise, we noticed \( e^{x^2} \) and \( 2x \) paired together. This suggested using the substitution \( u = e^{x^2} \), which aligned with our function's derivative. Consequently, \( du = 2x e^{x^2} \ dx \).
This substitution was successful because it led us to a much simpler integral form. We transitioned from variables like \( x \) to \( u \) by:

• Changing the function
• Adjusting the differential from \( dx \) to \( du \)
• Transforming the limits accordingly

The substitution method often requires translating limits of integration from the original variable to the new one, like changing from \( x=0 \) to \( x=\sqrt{\ln \pi} \) into \( u=1 \) to \( u=\pi \). This adaptability is key to resolving complex integrals efficiently.

Exponential Functions

Exponential functions appear frequently in calculus, particularly in integration problems due to their distinctive and often complex nature. Integrating exponential functions requires an understanding of their basic forms and properties.
In this exercise, encountering \( e^{x^2} \) inside the integral was a crucial aspect. It shaped our choice of substitution, transforming our problem.
Exponential functions, like \( e^{x} \), have straightforward derivatives and integrals, which, however, become intricate when combined with other functions. The power \( x^2 \) within the exponent necessitated careful handling using substitution.
When dealing with exponential functions, remember:

• The exponential function \( e^x \) is unique because its derivative is itself.
• Substitution can often transform an exponential function into a simpler form for integration.
• Changes in variables or limits are linked intricately with exponential growth or decay, impacting the integral's evaluation.

Understanding these essentials enables students to tackle diverse problems involving exponential and other complex functions.