Problem 25 - Calculus

Question

Use the method of substitution to find each of the following indefinite integrals. $$ \int x \sin \left(x^{2}+4\right) d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( -\frac{1}{2} \cos(x^2 + 4) + C \).

Step 1: Identify the Substitution

To solve \( \int x \sin(x^2 + 4) \, dx \), choose a substitution that simplifies the integral. Let \( u = x^2 + 4 \). This substitution is chosen because the derivative \( du = 2x \, dx \) relates to the \( x \, dx \) term in the integral.

Step 2: Solve for dx

From the substitution \( u = x^2 + 4 \), differentiate both sides with respect to \( x \) to find \( du \). This gives \( du = 2x \, dx \). Solve for \( dx \) to get \( x \, dx = \frac{1}{2} du \).

Step 3: Substitute and Simplify the Integral

Substitute \( u = x^2 + 4 \) and \( x \, dx = \frac{1}{2} du \) into the integral. This transforms the original integral into:\[ \int x \sin(x^2 + 4) \, dx = \int \frac{1}{2} \sin(u) \, du \]

Step 4: Integrate with Respect to u

Now, integrate \( \frac{1}{2} \sin(u) \) with respect to \( u \). The integral of \( \sin(u) \, du \) is \( -\cos(u) + C \), so we have:\[ \int \frac{1}{2} \sin(u) \, du = -\frac{1}{2} \cos(u) + C \]

Step 5: Substitute Back to Original Variable

Replace \( u \) with \( x^2 + 4 \) to express the antiderivative in terms of \( x \):\[ -\frac{1}{2} \cos(x^2 + 4) + C \]

Key Concepts

Indefinite Integrals

Indefinite integrals are a fundamental concept in calculus. They represent the collection of all antiderivatives of a function, which means they show all functions whose derivative equals the original function. In indefinite integrals, we are continuously seeking the general formula without any specific limits of integration.
An integral is represented by the symbol \( \int \) followed by the integrand, which is the function we want to integrate. In this case, the problem presents \( \int x \sin(x^2 + 4) \, dx \). The result of an indefinite integral also includes a constant of integration, denoted by \( C \), because integration is the reverse process of differentiation and constants disappear when differentiating.

• The aim of the indefinite integral is to find a function whose derivative is the given integrand.
• The integral sign \( \int \) and \( dx \) indicate the variables of integration.
• The integration constant \( C \) accounts for any constant term that differentiated to zero.

Understanding indefinite integrals is crucial, as they form the basis for solving many real-world problems, especially in physics and engineering, where functions model continuous change.

Trigonometric Functions

Trigonometric functions are essential in calculus as they model periodic phenomena. The main trigonometric functions—sine, cosine, and tangent—often occur in calculus problems and solutions. In this exercise, the function \( \sin(u) \) plays a central role. It simplifies the integration process after substitution.
These functions relate angles of a right triangle to the ratios of two side lengths and extend naturally to describe circular, oscillatory motion. Here are few things to note about trigonometric functions in calculus contexts:

• The integration of \( \sin(x) \) results in \(-\cos(x)\), as shown in the solution step 4 \( -\frac{1}{2} \cos(u) + C \).
• Understanding how to integrate and differentiate trigonometric functions is crucial, as they have unique behaviors that deviate from polynomial functions.

Knowing how to work with these functions broadens your calculus toolbox, allowing you to handle a wider array of mathematical challenges.

Integration Techniques

To tackle complicated integrals, several integration techniques simplify the task. The substitution method, a powerful technique showcased in this exercise, transforms a challenging integral into a more manageable one.
The substitution method involves replacing a complicated expression with a simpler one. By letting \( u = x^2 + 4 \), and thus finding \( du = 2x \, dx \), the original integral becomes easier to handle. Here's how it generally works:

• Select a substitution function \( u \) such that its derivative \( du \) is related to the non-integral part of the integrand—here, \( x \, dx \) matches \( \frac{1}{2} du \).
• Transform the variables, rewrite the integral completely in terms of \( u \), simplifying it.
• Integrate the new, simpler integral in terms of \( u \).
• Substitute back the original variables to get a solution relevant to the original integrand.

Integration techniques are vital for solving complex integrals appearing in higher mathematics and applied domains. The substitution method, in particular, is a foundational technique that saves time and effort.