Problem 59 - Applied Calculus

Question

Find the indefinite integrals. $$ \int x \cos \left(x^{2}+4\right) d x $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Integration Candidate and Substitution

The given integral is \( \int x \cos(x^2 + 4) \, dx \). Notice that the derivative of \( x^2 + 4 \) with respect to \( x \) is \( 2x \). This suggests using substitution. Let \( u = x^2 + 4 \).

Step 2: Calculate Derivative for Substitution

Differentiate \( u = x^2 + 4 \) with respect to \( x \) to find \( du \). We get \( \frac{du}{dx} = 2x \), or \( du = 2x \, dx \). This can be rearranged to \( \frac{du}{2} = x \, dx \).

Step 3: Substitute Variables in Integral

Substitute \( u = x^2 + 4 \) and \( x \, dx = \frac{du}{2} \) into the integral. The integral becomes \( \int \cos(u) \cdot \frac{1}{2} \, du \), which simplifies to \( \frac{1}{2} \int \cos(u) \, du \).

Step 4: Integrate with Respect to Substituted Variable

Integrate with respect to \( u \). The integral \( \int \cos(u) \, du \) is equal to \( \sin(u) + C \). Therefore, \( \frac{1}{2} \int \cos(u) \, du = \frac{1}{2} \sin(u) + C \).

Step 5: Substitute Back to the Original Variable

Replace \( u \) back with \( x^2 + 4 \) to obtain the final answer. The integrated function in terms of \( x \) is \( \frac{1}{2} \sin(x^2 + 4) + C \).

Key Concepts

Substitution Method

The substitution method is a technique used in calculus to simplify the process of finding integrals. It involves replacing a complicating expression within an integral with a simpler variable, making the integration easier. In the given exercise, we have the integral \( \int x \cos(x^2 + 4) \, dx \). The presence of \( x^2 + 4 \) prompts the use of substitution, because its derivative is proportional to \( x \, dx \), which is already present.

• Start by identifying parts of the integral that make substitution useful, commonly expressions that can be differentiated to produce parts already within the integral.
• For our problem, let \( u = x^2 + 4 \). This substitution simplifies the integral by transforming potential cumbersome expressions into easier ones.

This step significantly reduces the complexity of the integral.

Integration by Substitution

Integration by substitution is akin to the reverse process of the chain rule used in differentiation. It's one of the fundamental tools for solving integrals where a direct approach is not straightforward. In the problem given, after identifying \( u = x^2 + 4 \), you need to find \( du \) in terms of \( dx \).

• Differentiate \( u \) with respect to \( x \) to get \( du = 2x \, dx \). Rearrange this to \( x \, dx = \frac{du}{2} \).
• Substitute these expressions into the integral, converting it to \( \int \cos(u) \cdot \frac{1}{2} \, du \).

Now, you integrate with respect to the new variable \( u \), which is often more straightforward than the original expression.

Trigonometric Integration

Trigonometric integration involves integrals containing trigonometric functions, which often require specific techniques to solve. Once the integral is converted using substitution into \( \frac{1}{2} \int \cos(u) \, du \), it becomes clear that this is a form of trigonometric integration.

• The integral \( \int \cos(u) \, du \) yields \( \sin(u) + C \), utilizing the basic integral of the cosine function.
• Applying the constant multiplier from earlier, we have \( \frac{1}{2}(\sin(u) + C) \).

Finally, reverting \( u \) back to terms of \( x \), we achieve the solution \( \frac{1}{2} \sin(x^2 + 4) + C \). This demonstrates how substitution can simplify trigonometric integrals, ultimately leading to a straightforward antiderivative.