Problem 28 - Calculus for Biology and Medicine

Question

Use substitution to evaluate the indefinite integrals. $$ \int \sec ^{2} x e^{\tan x} d x $$

Step-by-Step Solution

Answer

The integral is \( e^{\tan x} + C \).

Step 1: Identify a Suitable Substitution

Identify a function and its derivative within the integral that can be substituted. Notice that the derivative of \( \tan x \) is \( \sec^2 x \). This suggests that we can make the substitution \( u = \tan x \), which leads to \( \frac{du}{dx} = \sec^2 x \) and thus \( du = \sec^2 x \, dx \).

Step 2: Perform the Substitution

Substitute \( u = \tan x \) and \( du = \sec^2 x \, dx \) into the integral: \[ \int \sec^2 x \, e^{\tan x} \, dx = \int e^u \, du. \]

Step 3: Integrate with Respect to u

Integrate \( \int e^u \, du \). The integral of \( e^u \) with respect to \( u \) is simply \( e^u + C \), where \( C \) is the constant of integration.

Step 4: Substitute Back for x

Substitute \( u = \tan x \) back into the expression to express the integral in terms of \( x \): \[ \int e^u \, du = e^{\tan x} + C. \]

Key Concepts

Substitution Method

The substitution method is a powerful technique in calculus that simplifies the process of finding indefinite integrals. It involves changing variables to make an integral easier to solve. By recognizing a part of the integrand that can be substituted with a single variable, solving the integral becomes straightforward.

This method hinges on identifying a function and its derivative within the integral. Here, we notice that the derivative of \(\tan x\) is \(\sec^2 x\). This insight guides us to use the substitution \(u = \tan x\), with \(du = \sec^2 x \, dx\).

Once this change is made, the integral \(\int \sec^2 x \, e^{\tan x} \, dx\) transforms into \(\int e^u \, du\), simplifying the calculus operation significantly.

By practicing this method, you'll see how it not only makes the process more manageable but often uncovers the structures and symmetries in complex integrals.

Integration Techniques

Integration techniques offer a range of strategies to evaluate integrals that are not immediately solvable. Various approaches, including substitution, integration by parts, and partial fractions, are commonly used.

Substitution is perhaps the most intuitive among these techniques, as it transforms the integral into a simpler one that's easier to manage. For the integral \(\int \sec^2 x \, e^{\tan x} \, dx\), substitution allows us to convert it into the much simpler form \(\int e^u \, du\).

This particular substitution was chosen because \(\sec^2 x\) appears as a multiplicative factor, which is the derivative of \(\tan x\). By recasting the integral in terms of \(u\), we exploit the fact that the integral of \(e^u\) is straightforward, yielding \(e^u + C\).

Mastering various integration techniques is crucial for tackling a wide variety of calculus problems efficiently and is a vital skill for any aspiring mathematician.

Calculus Problem Solving

Effective calculus problem solving involves breaking down complex problems into manageable steps. This structured approach often involves identifying appropriate techniques and following through with them methodically.

For the integral at hand, the problem solving starts with recognizing the relationship between the functions \(\tan x\) and \(\sec^2 x\). This recognition makes substitution the logical first step, turning a complex problem into a manageable one.

Solving the integral \(\int \sec^2 x \, e^{\tan x} \, dx\), involves identifying \(u = \tan x\) and \(du = \sec^2 x \, dx\), performing the substitution, integrating \(e^u\), and substituting back to express the solution in terms of \(x\).

This methodical breakdown is a hallmark of problem solving in calculus, illustrating how large, seemingly complex problems are often more straightforward than they initially appear. By applying these principles, students can develop confidence and competence in navigating calculus.