Problem 24 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(23-32\). $$ \int e^{x} \sec ^{3} e^{x} d x $$

Step-by-Step Solution

Answer

\( \int e^x \sec^3(e^x) \, dx = \frac{1}{2}(\sec(e^x) \tan(e^x) + \ln|\sec(e^x) + \tan(e^x)|) + C \).

Step 1: Substitution

First, let's use substitution to simplify the integral. Set \( u = e^x \), which implies that \( du = e^x \, dx \) or \( dx = \frac{du}{e^x} = \frac{du}{u} \).

Step 2: Substituting into the Integral

Replace \( e^x \) with \( u \) and \( dx \) with \( \frac{du}{u} \). The integral becomes \( \int e^x \sec^3(e^x) \, dx = \int u \sec^3(u) \frac{du}{u} = \int \sec^3(u) \, du \).

Step 3: Integrating \( \sec^3(u) \)

The integral of \( \sec^3(u) \) can be solved using integration by parts. Let \( v = \sec(u) \tan(u) \) and \( dv = \sec(u)(\sec^2(u) + \tan^2(u)) \, du \). Use integration by parts \( \int v \, du = uv - \int u \, dv \).

Step 4: Integration by Parts

Apply integration by parts:- Choose \( f = \sec(u) \) and \( dg = \sec^2(u) \, du \).- Then \( df = \sec(u) \tan(u) \, du \) and \( g = \tan(u) \).- Therefore, \( \int \sec^3(u) \, du = \sec(u) \tan(u) - \int \sec(u) \tan^2(u) \, du \).

Step 5: Simplification

Notice that \( \sec(u) \tan^2(u) = \sec(u)(\sec^2(u) - 1) = \sec^3(u) - \sec(u) \).Therefore, \( \int \sec^3(u) \, du = \sec(u) \tan(u) - (\int \sec^3(u) \, du - \int \sec(u) \, du) \).

Step 6: Solving the Equation

Rearrange the equation:\( 2 \int \sec^3(u) \, du = \sec(u) \tan(u) + \ln|\sec(u) + \tan(u)| \).So, \( \int \sec^3(u) \, du = \frac{1}{2}(\sec(u) \tan(u) + \ln|\sec(u) + \tan(u)|) \).

Step 7: Substitute Back

Recall \( u = e^x \). Substitute back:\( \int e^x \sec^3(e^x) \, dx = \frac{1}{2}(\sec(e^x) \tan(e^x) + \ln|\sec(e^x) + \tan(e^x)|) + C \), where \( C \) is the constant of integration.

Key Concepts

Integral Calculus

Integral calculus is a key area of calculus concerned with the accumulation of quantities, such as areas under curves, given the function that describes the curve. The primary operation in integral calculus is integration, which finds the function from its derivative (or rate of change).
\(\)When approaching integration, there are two main types: indefinite and definite. Indefinite integrals do not have limits of integration, meaning the result is a general function plus a constant. In contrast, definite integrals have upper and lower limits and result in a number that represents the net area under the curve within that interval.
\(\)To better understand integration, think of it as the reverse of differentiation. While differentiation splits a whole into parts, integration combines parts to make a whole. This operation has a wide range of applications:

• Calculating areas and volumes
• Solving complex equations in physics
• Understanding motion through displacement

Definite Integrals

Definite integrals are a fundamental part of integral calculus. They are used to compute the accumulation of quantities within a specific interval. The definite integral of a function \( f(x) \) from \( a \) to \( b \) is denoted by:\[\int_{a}^{b} f(x) \, dx\]This expression equals the net area between the curve \( f(x) \) and the x-axis from \( x = a \) to \( x = b \).
\(\)Definite integrals resolve to a numerical value rather than a function. This value represents:

• The total accumulation of the function over the interval
• An application to real-world problems such as distance traveled over time
• Optimization in economics and resource allocation

The Fundamental Theorem of Calculus links definite integrals and derivatives, allowing us to evaluate the integral using antiderivatives. This simplifies the process and broadens their application across multiple disciplines.

Substitution Method

The substitution method is a technique used in integration when the integrand is complicated or not easily integrable. It involves substituting a part of the integrand with a new variable to simplify the integral. This new variable substitution helps to transform the integral into a form that is easier to integrate.
\(\)To apply the substitution method:

• Identify part of the integrand to substitute with a new variable \( u \)
• Express \( dx \) in terms of \( du \)
• Rewrite the integral with the new variable
• Perform the integration on the simplified expression
• Substitute back the original variable after integrating

For example, consider the integral \( \int e^x \sec^3(e^x) \, dx \). Substituting \( u = e^x \) helps transform the integral into \( \int \sec^3(u) \, du \), which is simpler to solve. After integrating, replacing \( u \) with \( e^x \) gives the final result.