Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

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

\(-\frac{1}{2} \cos(x^2 + 4) + C\)

Step 1: Choose a Substitution

Identify a substitution that will simplify the integration problem. For the integral \( \int x \sin (x^2 + 4) \, dx \), let's use the substitution \( u = x^2 + 4 \).

Step 2: Calculate the Derivative

Differentiate the substitution to express \( dx \) in terms of \( du \). Start from \( u = x^2 + 4 \) and differentiate both sides: \( \frac{du}{dx} = 2x \) or equivalently \( du = 2x \, dx \).

Step 3: Solve for dx

Rearrange the equation from Step 2 to find \( dx \) in terms of \( du \):\[ dx = \frac{du}{2x} \]

Step 4: Substitute into the Integral

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

Step 5: Integrate with Respect to u

Now integrate \( \frac{1}{2} \int \sin (u) \, du \): The integral of \( \sin(u) \) is \(-\cos(u)\), so:\[ \frac{1}{2} \int \sin (u) \, du = -\frac{1}{2} \cos(u) + C \] where \( C \) is the constant of integration.

Step 6: Back-Substitute to Original Variables

Replace \( u \) with the original expression \( u = x^2 + 4 \):\[ -\frac{1}{2} \cos(u) + C = -\frac{1}{2} \cos(x^2 + 4) + C \]

Key Concepts

Substitution Method

The substitution method is a game-changer when it comes to solving complex indefinite integrals. It transforms an integral into a simpler form, making it easier to solve. To effectively use this method, we identify a part of the integral to substitute with a new variable—a clever choice can simplify the integration process significantly.
This new variable, commonly denoted as \( u \), should be chosen so that its derivative, \( du \), is already present in some form in the integral.

• Step 1: Choose a substitution that will simplify the integral, as seen in our exercise where we used \( u = x^2 + 4 \).
• Step 2: Differentiate \( u \) to express \( dx \) in terms of \( du \), resulting in \( du = 2x \, dx \).
• Step 3: Rearrange to solve for \( dx \), giving us \( dx = \frac{du}{2x} \).

By performing these calculations, we've prepared the integral for the substitution, paving the way for simpler integration. Choosing the right substitution is key as it determines the complexity of the resulting integral.

Integration Techniques

Integration techniques provide the tools necessary to solve related or complex integrals in a more straightforward manner. Substitution is just one of these numerous techniques. Each integral might require a different approach, but understanding these methods is essential for tackling a myriad of integration problems.
In our example, substitution drastically simplified the problem. After substituting and expressing \( dx \) in terms of \( du \), the integral reduced from \( \int x \sin(x^2 + 4) \, dx \) to \( \frac{1}{2} \int \sin(u) \, du \).
This reduction shows how substitution can transform an intricate problem into a simple standard form. Some other common techniques you might encounter include:

• Integration by Parts: Useful for products of functions.
• Partial Fraction Decomposition: Effective for rational functions.
• Trigonometric Substitution: Beneficial when dealing with integrals involving square roots.

Choosing the right integration technique, like substitution for this exercise, makes solving integrals more manageable and often more intuitive.

Trigonometric Integrals

Trigonometric integrals are integrals that involve trigonometric functions like sine, cosine, tangent, and others. Solving these requires understanding the properties and integral formulas of these functions.
In our exercise, the integral involves the function \( \sin(u) \), and its solution is based on the standard integrals of trigonometric functions. Specifically:

• The integral \( \int \sin(u) \, du \) simplifies to \(-\cos(u) + C \).

The crucial part is recognizing how trigonometric identities or substitutions apply to transform the original integral into these standard forms. By substituting \( x^2 + 4 \) with \( u \), we could directly address the integral of \( \sin(u) \).
Understanding these principles helps manage trigonometric integrals effectively. It’s also important to remember the constants and boundaries when handling definite integrals involving trigonometric forms on similar exercises.