Problem 117 - Calculus for Biology and Medicine

Question

Evaluate the definite integrals. $$ \int_{0}^{\pi / 2} x^{2} \exp \left(x^{3}\right) d x $$

Step-by-Step Solution

Answer

\( \frac{1}{3} ( \exp((\pi/2)^3) - 1 ) \) is the value of the integral.

Step 1: Identify the Integration Technique

The given integral \( \int_{0}^{\pi / 2} x^{2} \exp(x^3) \, dx \) suggests that a substitution method could be useful, particularly due to the form \( x^2 \cdot \exp(x^3) \). We can use substitution to simplify it.

Step 2: Perform a Substitution

Let \( u = x^3 \), which implies that \( du = 3x^2 \, dx \) or \( \frac{1}{3} \, du = x^2 \, dx \). Substitute \( u \) and \( x^2 \, dx \) into the integral, changing the limits of integration accordingly: when \( x = 0 \), \( u = 0^3 = 0 \), and when \( x = \pi/2 \), \( u = (\pi/2)^3 \).

Step 3: Simplify the Integral

Substituting into the integral, we have \( \int_{0}^{(\pi/2)^3} \frac{1}{3} \exp(u) \, du \). This simplifies to \( \frac{1}{3} \int_{0}^{(\pi/2)^3} \exp(u) \, du \).

Step 4: Evaluate the Exponential Integral

The integral of \( \exp(u) \) is simply \( \exp(u) \). Thus, evaluate from 0 to \( (\pi/2)^3 \): \( \frac{1}{3} \left[ \exp(u) \right]_0^{(\pi/2)^3} = \frac{1}{3} \left( \exp((\pi/2)^3) - \exp(0) \right) \).

Step 5: Simplify and Compute the Result

Compute the values at the limits: \( \exp(0) = 1 \), and the expression becomes \( \frac{1}{3} ( \exp((\pi/2)^3) - 1 ) \). This is the final evaluated result of the integral.

Key Concepts

Understanding Integration Techniques

Integration is a fundamental concept in calculus used to find areas under curves, among many other applications. The correct choice of technique often simplifies complex integrals. For definite integrals, like the one in the given exercise \( \int_{0}^{\pi / 2} x^{2} \exp(x^3) \ dx \), choosing the right technique is crucial for efficient solving. Here are some common integration techniques:

• **Substitution Method:** Useful when the integral contains a function and its derivative.
• **Integration by Parts:** Helps integrate products of functions, following a formula derived from the product rule.
• **Partial Fraction Decomposition:** Breaks down complex rational functions into simpler partial fractions.

Each technique has its own application scenarios. In our problem, due to the presence of \( x^2 \cdot \exp(x^3) \), substitution is preferable as it simplifies the integration process. When confronted with integrals that resemble a derivative of a composed function, substitution should often be your first tool.

Using the Substitution Method

The substitution method is an integration technique where we replace a part of the integrand with a single variable. This helps simplify an integral to a more manageable form. In the exercise, we had an integral \( x^2 \exp(x^3) \). Here’s how substitution helps:

• **Step 1:** Identify a suitable substitution. Look for a function whose derivative is also present within the integrand. For \( x^2 \exp(x^3) \), setting \( u = x^3 \) is effective because the differential \( du = 3x^2 \, dx \) fits nicely.
• **Step 2:** Solve for \( dx \). Express \( dx \) in terms of \( du \), which results as \( \frac{1}{3} \, du = x^2 \, dx \).
• **Step 3:** Substitute in the integral. Replace \( x^2 \, dx \) with \( \frac{1}{3} \, du \) and update the limits of integration based on \( u \). This changes the integral to a simpler form: \( \frac{1}{3} \int_{0}^{(\pi/2)^3} \exp(u) \, du \).

By making the right substitution, we transform the original integral into an easier-to-compute exponential integral.

Evaluating the Exponential Integral

An exponential integral involves the integration of functions of the form \( \exp(u) \), which signifies a straightforward integration process. When we have an integral like \( \int \exp(u) \, du \), it directly results in \( \exp(u) + C \), where \( C \) is the constant of integration. For definite integrals, the process includes evaluating this antiderivative at the given limits. In the exercise, after substitution, the integral became \( \frac{1}{3} \int_{0}^{(\pi/2)^3} \exp(u) \, du \), simplified further as follows:

• **Compute the antiderivative:** Simply \( \exp(u) \).
• **Evaluate the definite integral:** Use the limits of integration. So, substitute back: \( \frac{1}{3} [\exp(u)]_{0}^{(\pi/2)^3} \).
• **Final Calculations:** Solve \( \frac{1}{3} (\exp((\pi/2)^3) - \exp(0)) \). With \( \exp(0) = 1 \), you have your final evaluated result as \( \frac{1}{3} (\exp((\pi/2)^3) - 1) \).

By understanding the simplicity of exponential function integration and how limits work, you can evaluate even complex-looking integrals with confidence.