Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{1}\left(x^{2}+1\right)^{10}(2 x) d x $$

Step-by-Step Solution

Answer

The value of the definite integral is \(\frac{2047}{11}\).

Step 1: Identify the Inner Function for Substitution

Let's choose the inner function of the composite function as our substitution variable. In this case, the inner function is \(u = x^2 + 1\).

Step 2: Differentiate the Substitution Variable

Differentiate \(u = x^2 + 1\) with respect to \(x\). This yields \(\frac{du}{dx} = 2x\). Therefore, \(du = 2x \cdot dx\).

Step 3: Change the Limits of Integration

Since \(u = x^2 + 1\), substitute the limits of integration. When \(x = 0\), \(u = 0^2 + 1 = 1\), and when \(x = 1\), \(u = 1^2 + 1 = 2\). Thus, the new limits of integration for \(u\) are from 1 to 2.

Step 4: Perform the Integration in Terms of \(u\)

Rewrite the integral in terms of \(u\) and \(du\):\[\int_{1}^{2} u^{10} \, du\].Now, use the power rule for integration: \(\int u^n \, du = \frac{u^{n+1}}{n+1} + C\). Thus, \[ \int u^{10} \, du = \frac{u^{11}}{11} \].

Step 5: Evaluate the Integral with New Limits

Now, evaluate \(\frac{u^{11}}{11}\) from 1 to 2:\[\left[\frac{2^{11}}{11} - \frac{1^{11}}{11}\right] = \frac{2048}{11} - \frac{1}{11} = \frac{2047}{11}.\]

Key Concepts

Inner Function Substitution

When solving integrals using the substitution rule, one of the first tasks is identifying the inner function. This refers to a function within another function (often called a composite function).
In our problem, the composite function is \( (x^2 + 1)^{10} \).
Recognizing \( x^2 + 1 \) as the inner function simplifies the integration process by setting it equal to a new variable \( u \).

• Identify the component of the function that can be isolated.
• It should be simple to differentiate.
• This variable substitution helps in transforming the integral into a simpler form.

Think of it as changing the perspective of the problem to make computation easier.

Differentiating Substitution Variable

After choosing the inner function, the next step is to differentiate it with respect to \( x \).
This means finding how changes in \( x \) affect \( u \).
For \( u = x^2 + 1 \), differentiating gives \( \frac{du}{dx} = 2x \).

• The result is simplified to \( du = 2x \, dx \)
• This calculation is crucial as it links \( du \)and \( dx \)
• Ensure the entire differential \( dx \)is expressed in terms of \( du \)

Essentially, you are finding the rate at which \( u \)changes as \( x \)changes, facilitating the conversion of the integral into the variable \( u \).

Changing Limits of Integration

Changing the limits of integration is an integral part of substitution. Since you're using a new variable, \( u \), the original limits in terms of \( x \)must also be converted.
In our example:

• When \( x = 0 \), compute \( u = 0^2 + 1 = 1 \)
• When \( x = 1 \), compute \( u = 1^2 + 1 = 2 \)

This means the new limits range from 1 to 2.
This step ensures that the integral evaluation bounds correspond with the transformed function, allowing correct computation within the new variable's domain.

Power Rule for Integration

The power rule for integration is a fundamental principle that allows you to integrate expressions of the form \( u^n \).
This rule states that \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \).Applying this rule:

• Use it on the integral \( \int_{1}^{2} u^{10} \, du \)
• Apply it yielding \( \frac{u^{11}}{11} \)
• No constant needed since we're evaluating definite integrals

Finally, calculate the definite integral from the updated limits, 1 to 2.
Plug in the limits to get the exact numerical result: \( \left[\frac{2^{11}}{11} - \frac{1^{11}}{11}\right] = \frac{2048}{11} - \frac{1}{11} = \frac{2047}{11} \).
Practicing the power rule ensures a streamlined process for integrating polynomials and similar power functions.