Problem 55 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(41-62\). $$ \int_{0}^{\pi / 4}\left(1+e^{\tan \theta}\right) \sec ^{2} \theta d \theta $$

Step-by-Step Solution

Answer

The integral evaluates to \(e\).

Step 1: Define the integral

The given integral is \( \int_{0}^{\pi / 4}\left(1+e^{\tan \theta}\right) \sec ^{2} \theta d \theta \). This integral will be evaluated over the interval from \( 0 \) to \( \pi/4 \).

Step 2: Identify the substitution

Recognize that the integrand contains \(\tan \theta\) and \(\sec^2 \theta\). Therefore, use the substitution \( u = \tan \theta \), leading to \( du = \sec^2 \theta \, d\theta \).

Step 3: Change limits of integration

Convert the limits of integration with respect to \(\theta\) to limits concerning \(u\). When \(\theta = 0\), \(u = \tan(0) = 0\), and when \(\theta = \pi/4\), \(u = \tan(\pi/4) = 1\).

Step 4: Substitute and simplify the integral

Rewrite the integral in terms of \(u\). Substituting into the integral gives: \( \int_{0}^{1} (1 + e^u) \, du \).

Step 5: Solve the integral

Evaluate the integral: \[ \int_{0}^{1} (1 + e^u) \, du = \int_{0}^{1} 1 \, du + \int_{0}^{1} e^u \, du. \] The first integral results in \([u]_0^1 = [1 - 0] = 1\) and the second integral results in \([e^u]_0^1 = [e^1 - e^0] = e - 1\).

Step 6: Compute the final result

Add the results from the separate integrals: \(1 + (e - 1) = e\). So, the value of the original integral is \(e\).

Key Concepts

Integration Techniques

Integration is a powerful tool in calculus that allows you to find areas under curves, solve differential equations, and more. There are various integration techniques, and it's important to choose the right one to simplify a problem.

• Substitution: This technique is useful when you can transform the integrand into a simpler form. By substituting variables, you can convert a complex integral into a more manageable one.
• Integration by Parts: Based on the product rule for differentiation, this technique is suitable for products of functions.
• Partial Fractions: This approach decomposes a fraction into a sum of simpler fractions, allowing integration term by term.

In calculus, each integration technique has specific scenarios where it's most effective. In the example problem, the substitution method was applied because the integrand involved functions of \( \tan \theta \) and its derivative \( \sec^2 \theta \). This connection made substitution a natural and effective choice.

Definite Integrals

Definite integrals are used to calculate the area under a curve between two specified points on the x-axis. Unlike indefinite integrals, which yield a family of functions, definite integrals give a specific numerical value. The limits of integration, known as the upper and lower bounds, play a crucial role here.
In the given problem, the integral is evaluated from \( \theta = 0 \) to \( \theta = \pi/4 \). These boundaries define where the integral starts and ends on the curve. This process of integration over a specific interval results in a precise value, representing the exact area or accumulation of the values described by the integrand across that interval.
Remember:

• The Fundamental Theorem of Calculus connects derivatives and integrals, providing a way to evaluate definite integrals through the use of antiderivatives.
• A definite integral has geometric interpretation tied to the concept of area under a curve.
• Changing limits of integration is an essential step when using substitution methods, ensuring the integral is evaluated correctly within the new variable context.

Understanding the limits and the impact of the bounds is key to solving definite integrals effectively.

Substitution Method

The substitution method, often referred to as "u-substitution," is a technique used to simplify integrals by changing variables. This approach relies on finding an appropriate substitution that will make the integral easier to solve.
In simple terms:

• Identify a part of the integrand that can be substituted with a single variable, denoted usually as \( u \).
• Compute the derivative of this substitution to change the differential, noted as \( du \).
• Transform the integral into terms of \( u \) and \( du \), which often simplifies the expression.
• Don't forget to adjust the limits of integration if evaluating a definite integral. The new limits will correspond to the original function's limits transformed through your substitution.
• Finally, integrate with respect to \( u \), and if needed, substitute back to the original variable to find the solution.

In the example provided, the substitution \( u = \tan \theta \) was chosen because \( du = \sec^2 \theta \, d\theta \), perfectly matching a part of the integrand. This allowed the seamless transition from an integral in \( \theta \) to one in \( u \), making the problem significantly easier to resolve.
Mastering this technique widens your toolset when approaching difficult integrals, making seemingly complex problems often quite solvable.