Problem 57 - Calculus Early Transcendentals: Pearson New International Edition

Question

, use the Substitution Rule for Definite Integrals to evaluate each definite integral.$$ \int_{0}^{1} x \cos ^{3}\left(x^{2}\right) \sin \left(x^{2}\right) d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(-\frac{1}{8}(\cos^4(1) - 1)\).

Step 1: Identify the Inner Function

First, recognize that the expression inside the cosine and sine functions, \(x^2\), is likely to be our substitution variable. This is because it appears twice within the integral. Let's set \(u = x^2\).

Step 2: Differentiate the Substitution Variable

Differentiate \(u = x^2\) with respect to \(x\) to find \(du\): \(du = 2x \, dx\). This will help us replace \(dx\) in our integral.

Step 3: Solve for dx in terms of du

Rearrange the expression for \(du\) to solve for \(dx\): \(dx = \frac{1}{2x} \, du\).

Step 4: Change Limits of Integration

Substitute the original limits of integration in terms of \(x\) with \(u\). When \(x = 0\), \(u = 0^2 = 0\). When \(x = 1\), \(u = 1^2 = 1\). Thus, the limits remain the same, from 0 to 1.

Step 5: Substitute and Simplify the Integral

Substitute the expressions for \(u\), \(x \, dx\), and \(dx\) into the integral: \[\int_{0}^{1} x \, \cos^3(u) \, \sin(u) \cdot \frac{1}{2x} \, du = \frac{1}{2} \int_{0}^{1} \cos^3(u) \sin(u) \, du.\]The \(x\) terms cancel out, simplifying the integral.

Step 6: Execute the Integration

Find the integral of \(\cos^3(u) \sin(u)\): \[\int \cos^3(u) \sin(u) \, du = -\frac{1}{4} \cos^4(u) + C.\]Therefore, the definite integral becomes:\[-\frac{1}{8} \left[ \cos^4(u) \right]_0^1.\]

Step 7: Evaluate Definite Integral

Evaluate at the bounds 0 and 1:\[-\frac{1}{8}(\cos^4(1) - \cos^4(0)) = -\frac{1}{8} \left( \cos^4(1) - 1 \right).\]

Key Concepts

Definite Integration

Definite integration is a powerful method in calculus used to find the accumulation of quantities over an interval. It calculates the net area under a curve, considering specific upper and lower bounds. In this exercise, we have a definite integral where the limits are from 0 to 1. It's interpreted geometrically as the area beneath the function within this interval.

Key points to remember about definite integration include:

• Definite integrals produce a numerical result, not a function.
• They depend on the limits of integration, which indicate the interval over which you're integrating.
• They account for both areas above and below the x-axis, where areas beneath the axis contribute negatively.

Unlike indefinite integrals, these do not have a constant of integration since the calculation yields a finite number.

Trigonometric Integration

Trigonometric integration involves integrating functions that contain trigonometric functions like sine, cosine, and others. The integral in the original exercise features \( \cos^3(x^2) \) and \( \sin(x^2) \), which are trigonometric expressions.

When integrating trigonometric functions, some techniques that are often used include:

• Leveraging trigonometric identities to simplify expressions.
• Utilizing substitution to handle products of different trigonometric functions.
• Simplifying powers of sine and cosine by converting them to identities that are easier to integrate, whenever possible.

This particular integral uses both cosine and sine in a form that makes it apt for the substitution method, demonstrating how these functions can be systematically tackled in calculus.

Substitution Method

The substitution method is a strategy for simplifying integrals, which is particularly useful when dealing with composite functions. In the given problem, we recognize \( x^2 \) as a potential candidate for substitution because it appears inside both trigonometric functions.

Here’s how substitution works:

• Identify the substitution variable, in this case, \( u = x^2 \).
• Differentiate to find \( du = 2x \, dx \), and solve for \( dx \).
• Substitute \( u \) and the expression for \( dx \) back into the integral, which simplifies the problem.

This method typically transforms a complex-looking integral into a simpler one, which is easier to evaluate. It hinges on the technique of altering the variable of integration to streamline the integral, ultimately leading to an easier computing task.

Calculus Problem Solving

Solving calculus problems often involves multiple strategies and logical reasoning. In this particular problem, we employ several techniques to resolve the integral efficiently.

Achieving a solution involves:

• Understanding the function and its components fully before integrating.
• Choosing appropriate methods — in this case, substitution simplifies the integration of trigonometric functions.
• Breaking problems into manageable steps, such as changing limits of integration and simplifying substitutions.
• Applying formulas and theorems correctly, like the integration of \( \cos^3(u)\sin(u) \).

Each step should progress logically from identifying the problem to realizing a numeric result, like \(-\frac{1}{8}(\cos^4(1) - 1)\) in this problem. This systematic approach is key to tackling complex calculus problems with confidence.