Problem 30 - Essential Calculus

Question

Think About It Explain why the two integrals are equal. \(\int_{1}^{e} \sqrt{1+\frac{1}{x^{2}}} d x=\int_{0}^{1} \sqrt{1+e^{2 x}} d x\) Use the integration capabilities of a graphing utility to verify that the integrals are equal.

Step-by-Step Solution

Answer

The two integrals are equal because, on applying the substitution \(u = e^{x}\), the integrand of the second integral becomes exactly the same as the first one, over the same interval [1, e]. Thus the two integrals are equal.

Step 1: Recognise the integral structure.

Notice that the second integral has a slightly more complex structure due to the exponential function. However, both functions share a common denominator under the square root. This similarity might hint at a transformation of variables we can use to demonstrate the equality.

Step 2: Decide on the set up for variable substitution.

In order to make the right hand side (RHS) integral look like the left hand side (LHS), let's substitute \(u = e^{x}\) in the RHS integral, which entails \(x = \ln u\) and \(dx = 1/u \, du\). Note that as \(x\) ranges from \(0\) to \(1\), \(u\) ranges from \(1\) to \(e\).

Step 3: Carry out the variable substitution.

Substitute the new variables into the RHS integral: \[\int_{1}^{e} \sqrt{1 + u^{2}} \cdot \frac{1}{u} \, du\] Notice that after simplifying, this is exactly the same as the LHS integral.

Key Concepts

Integration techniques

When tackling definite integrals, it helps to have a strong grasp of various integration techniques. They allow us to simplify and solve integrals more effectively. One common approach is substitution, which can help you recognize and transform complex expressions. The key is to identify patterns that can be altered into more manageable forms. By restructuring an integral using substitution, we can reveal hidden equivalences, making computations easier to handle. Always take note of the limits of integration as you perform this transformation, to ensure accurate calculations.

Variable substitution

Variable substitution is a powerful method used in integration to change the variables in a function, making it simpler to integrate. In the context of our problem, substituting the variable lets us show that two different integrals are equivalent.

• Substitute a new variable for a specific expression in the integrand.
• Adjust the differential and limits of integration to match the substitution.

For example, we substituted \( u = e^x \), meaning \( dx = \frac{1}{u} du \). This transforms our integral, simplifying the calculation and demonstrating equivalency. It's important to carefully choose substitutions that streamline the function, allowing the integral to match another form.

Integral equivalence

To prove the equivalence of two integrals, observe their structural similarities. In our exercise, we looked at both integrals and identified a root function combined with common factors. Variable substitution, as used earlier, can demonstrate equivalence clearly. Here’s a quick checklist to verify integral equivalence:

• Conduct a thorough comparison of both integral structures.
• Perform variable substitution only when concrete correspondences between variables are identified.
• Confirm that the limits of integration adjust correctly as per the substitution process.

Ultimately, both integrals simplify to the same form, confirming their equality. Be sure to maintain attention to detail, as small errors might lead to incorrect equivalences.

Exponential functions

Exponential functions often appear in definite integrals, introducing complexity due to their growth behavior and transformation properties. Recognizing when these functions can be simplified can significantly aid in solving integrals effectively.

• Understand the general form: \( a^{x} \), with special focus on \( e^{x} \), the natural exponential function.
• Examine how exponential expressions can transform other components of the integral, such as changing logarithmic expressions when inverse operations are performed.

In the exercise, we ingeniously used the exponential function \( e^x \) to simplify the original setup, leading to an equivalent expression. Always look for these opportunities when handling exponential functions, as they can be the key to unlocking easier solutions.