Problem 20 - Applied Calculus

Question

Find the integrals .Check your answers by differentiation. $$\int 12 x^{2} \cos \left(x^{3}\right) d x$$

Step-by-Step Solution

Answer

The integral is \( 4 \sin(x^3) + C \).

Step 1: Identify Integration Technique

To find the integral \( \int 12x^2 \cos(x^3) \, dx \), look for a substitution that will simplify the integrand. Notice that the derivative of \( x^3 \) is \( 3x^2 \), which suggests using the substitution method.

Step 2: Perform Substitution

Let \( u = x^3 \). Then \( du = 3x^2 \, dx \) implies \( dx = \frac{du}{3x^2} \). Substitute back into the integral:\[ \int 12x^2 \cos(x^3) \, dx = \int 12x^2 \cos(u) \frac{du}{3x^2} \] Simplifying, we get:\[ \int 4 \cos(u) \, du \]

Step 3: Integrate the Simplified Integral

The integral \( \int 4 \cos(u) \, du \) can be solved directly:\[ \int 4 \cos(u) \, du = 4 \sin(u) + C \] where \( C \) is the constant of integration.

Step 4: Substitute Back in Terms of x

Replace \( u \) with \( x^3 \) to revert the substitution:\[ 4 \sin(x^3) + C \]

Step 5: Check the Solution by Differentiation

Differentiate the result \( 4 \sin(x^3) + C \) with respect to \( x \) to verify the solution:\[ \frac{d}{dx}[4 \sin(x^3) + C] = 4 \cos(x^3) \cdot \frac{d}{dx}[x^3] = 12x^2 \cos(x^3) \] This matches the original integrand, confirming the solution is correct.

Key Concepts

Substitution Method

The substitution method is a powerful technique used in calculus to simplify integrals. It involves changing variables to simplify the integration process. In the given problem, we need to find the integral \(\int 12x^2 \cos(x^3) \, dx\). Firstly, we identify the inner function \(x^3\), whose derivative is \(3x^2\). This gives a hint that substituting could help. By letting \(u = x^3\), we get \(du = 3x^2 \, dx\). We can rearrange this to find \(dx = \frac{du}{3x^2}\). When we substitute these into the integral, the \(x^2\) terms cancel out:

• Replace \(x^3\) with \(u\), simplifying \( \int 12x^2 \cos(u) \frac{du}{3x^2} \)
• The terms simplify to \( \int 4 \cos(u) \, du \)

This approach not only reduces the complexity of the integral but also reveals an integral that is straightforward to solve.

Trigonometric Integrals

Trigonometric integrals require us to integrate functions involving trigonometric identities. They often appear more complicated than they are. In our example, our integral was transformed to \(\int 4 \cos(u) \, du\). Integrating cosine involves finding a function whose derivative is cosine. Therefore, \(\int \cos(u) \, du = \sin(u) + C\).

• In this case, multiplying by 4 gives: \(\int 4 \cos(u) \, du = 4 \sin(u) + C\)
• After integrating, we cannot forget the constant of integration \(C\), which represents the family of solutions.

These steps provide essential practice in recognizing and computing basic trigonometric integrals, which are foundational in more advanced calculus applications.

Checking Solutions by Differentiation

Verifying integration by differentiating reverses the original process and helps to ensure that our solution is correct. Once we have obtained \(4 \sin(x^3) + C\) from integration, we differentiate this expression to check our work.Differentiating \(4 \sin(x^3) + C\) requires applying the chain rule:

• The derivative of \(\sin(x^3)\) with respect to \(x\) is \(\cos(x^3) \cdot \frac{d}{dx}[x^3] = 3x^2 \cos(x^3)\)
• Therefore, the derivative of \(4 \sin(x^3)\) is \(4 \cdot 3x^2 \cos(x^3)\)

This simplifies to \(12x^2 \cos(x^3)\), matching the original integrand and confirming the solution's correctness. Differentiation verification is a reliable method to verify results and deepen understanding of integration.