Problem 31 - Applied Calculus

Question

Find the integrals .Check your answers by differentiation. $$\int x \sin \left(4 x^{2}\right) d x$$

Step-by-Step Solution

Answer

The integral is \(-\frac{1}{8} \cos(4x^2) + C\).

Step 1: Recognize the Integral Type

The given integral \( \int x \sin(4x^2) \, dx \) is a substitution type integral because it contains a composite function \( \sin(4x^2) \). We can use substitution to simplify it.

Step 2: Choose an Appropriate Substitution

Let's set \( u = 4x^2 \). Thus, \( du = 8x \, dx \). Rewrite \( dx \) in terms of \( du \) and \( x \): \( dx = \frac{du}{8x} \).

Step 3: Substitute and Simplify

Substitute \( u = 4x^2 \) into the integral. This gives us:\[\int x \sin(u) \frac{du}{8x}\]Simplifying it, we have:\[\frac{1}{8} \int \sin(u) \, du\]

Step 4: Integrate

Now, integrate \( \sin(u) \, du \):\[\int \sin(u) \, du = -\cos(u)\]Thus, the integral becomes:\[-\frac{1}{8} \cos(u) + C\]

Step 5: Back-Substitute

Replace \( u \) with \( 4x^2 \) to express the integral in terms of \( x \):\[-\frac{1}{8} \cos(4x^2) + C\]

Step 6: Check by Differentiation

Differentiate the result to verify the integral. Compute:\[\frac{d}{dx} \left(-\frac{1}{8} \cos(4x^2) + C \right)\]Using the chain rule, this becomes:\[\frac{1}{8} \cdot \sin(4x^2) \cdot 8x = x \sin(4x^2)\]This matches the original integrand, confirming our solution is correct.

Key Concepts

Substitution Method

The substitution method is a popular technique in integral calculus used to make integrals simpler by transforming them into a more familiar form. It is particularly useful when dealing with composite functions.

• Recognize Composite Functions: Look for functions nested within other functions. In the integral \( \int x \sin(4x^2) \, dx \), \( \sin(4x^2) \) is a composite function, indicating that substitution might be useful.


• Choose a Substitution: Identify an expression to replace. Typically, this choice simplifies the differential. Here, we set \( u = 4x^2 \), transforming the expression to match typical integral forms.


• Rewrite the Differential: Calculate \( du \) in terms of \( dx \) to completely convert the integral variables. Here, \( du = 8x \, dx \), so \( dx = \frac{du}{8x} \).


• Substitute and Simplify: Replace variables in the integral, reducing complexity. Our substitution transforms the integral into \( \frac{1}{8} \int \sin(u) \, du \).

Chain Rule

The chain rule is a fundamental rule in differentiation used when differentiating composite functions. It simplifies computations when the function has another function inside it.

• Understanding the Rule: The chain rule states that if a function \( f \) is composed with another function \( g \), you'll differentiate \( f \) with respect to \( g \) and multiply by the derivative of \( g \). Mathematically, if \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).


• Application: In verifying our integral result, we employ the chain rule. When differentiating \( -\frac{1}{8} \cos(4x^2) \), recognize \( \cos(4x^2) \) is composed with \( 4x^2 \), requiring the chain rule. Thus, \( \frac{d}{dx} (-\frac{1}{8} \cos(4x^2)) = \frac{1}{8} \cdot 8x \cdot \sin(4x^2) \).


• Why It's Powerful: The chain rule makes dealing with nested functions systematic, ensuring correct differentiation routines even for complex expressions.

Differentiation

Differentiation is a crucial process in calculus used to compute the rate at which a function changes. It is the inverse operation of integration, and a method to verify the correctness of an integral.

• Verify Integrals: After computing an integral, differentiation helps check if the antiderivative is correct. Differentiating \(-\frac{1}{8} \cos(4x^2) + C\) yields the original integrand, confirming its correctness.


• Basic Rules: Differentiation involves basic rules like the power rule, product rule, and as noted earlier, the chain rule for composite functions. Together, they form the foundation of finding derivatives accurately.


• The Relationship with Integration: Differentiation and integration are interconnected. The Fundamental Theorem of Calculus ties both processes, rendering one the inverse of the other. Hence, any integral solved can be checked by differentiation.