Problem 26 - Calculus for Biology and Medicine

Question

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int \cos x e^{-\sin x} d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( -e^{-\sin x} + C \).

Step 1: Identify the Substitution

To evaluate the integral \( \int \cos x e^{-\sin x} \, dx \) using substitution, first look for a function and its derivative in the integrand. Here, the derivative of \( -\sin x \) is \( -\cos x \), which is present in the integral. So, let's set \( u = -\sin x \).

Step 2: Differentiate the Substitution

Differentiate \( u = -\sin x \) with respect to \( x \): \( \frac{du}{dx} = -\cos x \). Rearrange this to solve for \( dx \): \( du = -\cos x \, dx \), so \( dx = \frac{du}{-\cos x} \).

Step 3: Substitute in the Integral

Substitute \( u = -\sin x \) and \( dx = \frac{du}{-\cos x} \) into the original integral: \[ \int \cos x \, e^{-\sin x} \, dx = \int e^{u} \, (-\frac{du}{\cos x}) = \int - e^u \, du \].

Step 4: Evaluate the Integral

The integral \( \int - e^{u} \, du \) is \( - e^{u} + C \), where \( C \) is the constant of integration.

Step 5: Back-Substitute to Original Variable

Substitute back \( u = -\sin x \) into the expression: \[ - e^{u} + C = - e^{(-\sin x)} + C \].

Step 6: Simplify the Result

The simplified expression for the indefinite integral is \[ - e^{(-\sin x)} + C \], or \[ - e^{-\sin x} + C \].

Key Concepts

Understanding Indefinite Integrals

An indefinite integral represents a family of functions whose derivative matches the given function. This is different from definite integrals, where you calculate the area under a curve between two limits. The integral sign followed by a function, such as \( \int f(x) \, dx \), denotes an indefinite integral. When solving indefinite integrals, always include the constant of integration \( C \). This constant captures the idea that there are infinitely many antiderivatives for any function.

For example, if \( F(x) \) is an antiderivative of \( f(x) \), then \( F(x) + C \) is the general solution to the indefinite integral \( \int f(x) \, dx \).

• Includes a constant \( C \) symbolizing all potential solutions.
• No specific limits of integration are used.

Understanding this concept is essential because it anchors much of what you do in calculus when dealing with integrals. Always remember that indefinite integral solutions should include this constant to reflect all possible shiftings of the antiderivative graph.

Basics of Trigonometric Substitution

Trigonometric substitution is an integral solving technique used when an integrand involves specific forms that can be simplified using trigonometric identities. It is especially helpful when you encounter integrals containing expressions like \( \sqrt{a^2 - x^2} \), \( a^2 + x^2 \), or \( x^2 - a^2 \). These can be transformed using sine, cosine, or tangent substitutions due to their geometric relationships with right triangles.

In this exercise, no direct trigonometric substitution was made, but understanding this technique can deepen your integration skills when handling more complex integrals. Here's what to remember:

• Identify the standard form related to trigonometric identities.
• Replace expressions with trigonometric identities (like \( x = a \sin \theta \) for \( \sqrt{a^2 - x^2} \)).
• Solve the integral, then revert to the original variable.

Even if not directly applicable here, familiarity with trigonometric substitution prepares you for tackling tougher integrals involving composite functions and radicals.

Introduction to Integration Techniques

Integration techniques are essential tools in solving various types of integrals. Different techniques allow for the simplification of complex integrals through creative manipulation. Some common integration methods include substitution, integration by parts, partial fraction decomposition, and trigonometric substitution.

In the given solution, substitution was aptly chosen. This technique cleverly simplifies the process by introducing a new variable \( u \) for part of the integrand. It often turns a complex-looking integral into a solvable one. Here’s a quick guide to the substitution technique:

• Identify a portion of the integrand to substitute with a new variable \( u \).
• Ensure \( du \), the differential of \( u \), accounts for all parts of the integrand you're replacing.
• Substitute back the original variable after integration.

Each integration technique has its own situational best practices, and mastering them allows you to approach almost any integral with confidence. Remember, the choice of technique can drastically simplify your work or make an otherwise impossible integral possible.