Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{1}\left(x^{2}+1\right)^{10}(2 x) d x \quad\)

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{2047}{11}\).

Step 1: Identify the Substitution

In the given integral, we have \( \int_{0}^{1} (x^2 + 1)^{10} (2x) \, dx \). Notice that the derivative of \( x^2 + 1 \) is \( 2x \). This suggests that we use the substitution \( u = x^2 + 1 \).

Step 2: Express \( dx \) in Terms of \( du \)

Differentiate the substitution equation: \( du = 2x \, dx \). Therefore, \( dx = \frac{du}{2x} \).

Step 3: Change the Limits of Integration

When \( x = 0 \), \( u = 0^2 + 1 = 1 \). When \( x = 1 \), \( u = 1^2 + 1 = 2 \). Therefore, the limits of integration change from 0 to 1, to 1 to 2.

Step 4: Rewrite the Integral in Terms of \( u \)

Substitute \( u \) and \( dx \) into the integral: \[ \int_{1}^{2} u^{10} \, du \].

Step 5: Evaluate the Integral

Integrate \( u^{10} \) with respect to \( u \): \( \int u^{10} \, du = \frac{u^{11}}{11} + C \).

Step 6: Apply the Limits of Integration

Evaluate from 1 to 2: \( \left[ \frac{u^{11}}{11} \right]_{1}^{2} = \frac{2^{11}}{11} - \frac{1^{11}}{11} \).

Step 7: Simplify the Result

Calculate the values: \( \frac{2^{11}}{11} = \frac{2048}{11} \) and \( \frac{1^{11}}{11} = \frac{1}{11} \). The result is \( \frac{2048 - 1}{11} = \frac{2047}{11} \).

Key Concepts

Definite Integral

A definite integral is an important concept in calculus that helps us calculate the accumulated quantity, such as area, under a curve between two limits or points on the x-axis. When you see an integral with boundaries, like from 0 to 1, it indicates that the integral is definite. Unlike indefinite integrals, which yield general antiderivative functions, definite integrals involve boundaries, meaning you get a specific numerical value. This happens because, after integration, you substitute these limits into the antiderivative function and find the difference.

For instance, when evaluating \( \int_{0}^{1} (x^2 + 1)^{10} (2x) \, dx \), the computation leads to definite numerical results. Once the integral is evaluated and simplified, as shown in the original problem, it delivers the result \( \frac{2047}{11} \), representing the exact cumulative value of the function within those bounds.

Substitution Rule

The substitution rule is a technique used to simplify integrals and make them more manageable. It involves replacing a part of the integrand (the function to be integrated) with a new variable, usually \( u \), to simplify the integral. This is particularly useful when recognizing a function and its derivative within an integral.

Consider the integral \( \int_{0}^{1} (x^2 + 1)^{10} (2x) \, dx \). By identifying that the derivative of \( x^2 + 1 \) is \( 2x \), you can let \( u = x^2 + 1 \), which renders the problem amenable to easier integration. The derivative \( du = 2x \, dx \) allows rewriting \( dx \) as \( \frac{du}{2x} \). Thus, substituting makes the integral simpler: it changes variables and adjusts limits from \( 0 \) to \( 1 \) in the \( x \)-domain to \( 1 \) to \( 2 \) in the \( u \)-domain.

Integration Techniques

Integration techniques in calculus provide ways to solve complex integrals effectively. The substitution rule is one key method among various techniques employed to break down challenging problems. Other techniques include integration by parts, trigonometric substitution, and partial fraction decomposition.

The substitution rule is particularly handy when the integral involves a composition of functions where recognizing a portion of it as a derivative can drastically simplify the process. **How to identify when to use substitution?**

• Look for a portion of the integral that resembles a function's derivative.
• Determine if transforming the integral through substitution reduces its complexity.

Thus, transformation via substitution sets the scene for more straightforward computation by hiding evident complexities under a new, more convenient form.

Mathematical Computation

Mathematical computation in calculus refers to the process of performing calculations to derive a solution from integrals, derivatives, and other mathematical operations. The step-by-step approach aids in ensuring clarity and correctness in arriving at the answer.

When computing the definite integral \( \int_{0}^{1} (x^2 + 1)^{10} (2x) \, dx \), several steps are performed: identifying the substitution, changing variables, adjusting integration limits, evaluating the integral, and finally applying the limits of integration. This structured approach ensures that each part of a potentially complex problem is handled correctly.
**Why is computation necessary?**

• It confirms the understanding of mathematical principles.
• Ensures accuracy by methodically breaking down and solving integrals.
• Facilitates generalization to different problems with similar structures.

Thus, effective mathematical computation transforms theoretical calculus into practical results, like finding areas or analyzing functions over certain intervals.