Problem 58 - Calculus

Question

Use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{-\pi / 2}^{\pi / 2} x^{2} \sin ^{2}\left(x^{3}\right) \cos \left(x^{3}\right) d x $$

Step-by-Step Solution

Answer

The definite integral evaluates to \( \frac{2}{9} \sin^3(\frac{\pi^3}{8}) \).

Step 1: Identify the Substitution

We will use substitution to simplify the integration. Let's choose the substitution \( u = x^3 \). This implies that \( du = 3x^2 \, dx \), or \( dx = \frac{du}{3x^2} \).

Step 2: Change Limits of Integration

Since \( u = x^3 \), when \( x = -\frac{\pi}{2} \), \( u = \left(-\frac{\pi}{2}\right)^3 = -\frac{\pi^3}{8} \). When \( x = \frac{\pi}{2} \), \( u = \left(\frac{\pi}{2}\right)^3 = \frac{\pi^3}{8} \). So, the limits of integration change from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) for \( x \) to \( -\frac{\pi^3}{8} \) to \( \frac{\pi^3}{8} \) for \( u \).

Step 3: Substitute and Simplify the Integrand

Substitute \( u = x^3 \) into the integrand. We replace \( x^2 \sin^2(x^3) \cos(x^3) \) and \( dx \) as follows: \[ x^2 \sin^2(u) \cos(u) \frac{du}{3x^2} = \frac{1}{3} \sin^2(u) \cos(u) du \] Now the integral becomes: \[ \int_{-\frac{\pi^3}{8}}^{\frac{\pi^3}{8}} \frac{1}{3} \sin^2(u) \cos(u) \, du \]

Step 4: Evaluate the Simplified Integral

Factor out the constant \( \frac{1}{3} \): \[ \frac{1}{3} \int_{-\frac{\pi^3}{8}}^{\frac{\pi^3}{8}} \sin^2(u) \cos(u) \, du \] Now integrate using a new substitution if necessary. Here, let \( v = \sin(u) \), then \( dv = \cos(u) \, du \), and replace the limits accordingly: when \( u = -\frac{\pi^3}{8} \), \( v = \sin(-\frac{\pi^3}{8}) \); and when \( u = \frac{\pi^3}{8} \), \( v = \sin(\frac{\pi^3}{8}) \). The transformed integral becomes: \[ \frac{1}{3} \int_{\sin(-\frac{\pi^3}{8})}^{\sin(\frac{\pi^3}{8})} v^2 \ dv \]

Step 5: Integrate and Calculate

Integrate \( v^2 \): \[ \frac{1}{3} \left[ \frac{v^3}{3} \right]_{\sin(-\frac{\pi^3}{8})}^{\sin(\frac{\pi^3}{8})} \] Calculate the value: replace with limits and simplify \[ = \frac{1}{9} \left( \sin^3(\frac{\pi^3}{8}) - \sin^3(-\frac{\pi^3}{8}) \right) \] Since \( \sin(-a) = -\sin(a) \), you find that: \( \sin^3(\tilde u) - (-\sin^3(\tilde u)) = 2\sin^3(\tilde u) \), where \( \tilde u = \frac{\pi^3}{8} \). Thus: \[ = \frac{1}{9} \times 2\sin^3(\frac{\pi^3}{8}) = \frac{2}{9} \sin^3(\frac{\pi^3}{8}) \]

Step 6: Final Answer: Evaluate and Conclude

Calculate \( \sin^3(\frac{\pi^3}{8}) \) numerically if a specific value is needed. For this problem, the final solution is represented symbolically as \( \frac{2}{9} \sin^3(\frac{\pi^3}{8}) \).

Key Concepts

Integration Techniques

When tackling integrals, especially complex ones involving various functions, different integration techniques can simplify the process. Here, we focus on the **Substitution Rule**. This rule is akin to the chain rule in differentiation and is very useful when an integrand is a composition of functions.
The key idea of substitution is to transform a complex integral into a simpler one by replacing a part of the integrand with a new variable. For example, if an integral involves a function and its derivative, substitution can significantly reduce complexity.
In our exercise, we started with the substitution \( u = x^3 \). By changing the variable from \( x \) to \( u \), the problem became simpler to integrate. The substitution technique can make an otherwise daunting definite integral manageable.

Definite Integrals

Definite integrals are an essential aspect of calculus, representing the net area under a curve between two limits. These are different from indefinite integrals, which generally represent a family of antiderivatives without set limits.
To solve a definite integral using substitution, it is critical to change the limits of integration according to the new variable. For example, when we substitute \( u = x^3 \) in this exercise, we also need to update the limits from \( -\pi/2 \) and \( \pi/2 \) to \( -\pi^3/8 \) and \( \pi^3/8 \), respectively.
This step ensures that the transformed integral represents the same area under the curve as the original integral. Remember, the bounds also undergo transformation aligned with the substitution, which is key in successfully integrating within these limits.

Trigonometric Functions

Trigonometric functions like sine and cosine add a layer of complexity to integration. Their properties, like periodicity and symmetry, are often exploited to simplify integration.
In our problem, we have \( \sin^2\left(x^3\right) \) and \( \cos\left(x^3\right) \) inside the integral. Using trigonometric identities or substitutions can simplify such integrals. For instance, converting \( \sin^2(x) \) using identities makes it more manageable to integrate.
Within substitution steps, recognizing symmetrical limits like in this exercise, indicates potential further simplification. This is seen where \( \sin(-x) = -\sin(x) \) helps during calculations, providing symmetry used to evaluate the integral effectively.

Mathematical Substitution

Mathematical substitution simplifies complex integrals by changing the variable of integration to make the integral more straightforward.
The process involves selecting a substitution that usually simplifies a part of the integrand to align it with a standard form integral. For example, \( u = x^3 \) leads to simplifying \( x^2 \sin^2(u) \cos(u) \). Another substitution within this process was \( v = \sin(u) \), which easily aligns the integral into a simple polynomial form \( v^2 \).
Substitutions often require adjusting limits as seen in this step-by-step: change both the integrand and the integration limits to maintain the integrity of the integral. Overall, a strategic substitution reduces an elaborate calculus problem down to basic algebraic manipulation for effective integration.