/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 \(1-30=\) Evaluate the integral.... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 15

\(1-30=\) Evaluate the integral. $$\int_{0}^{1}\left(x^{10}+10^{x}\right) d x$$

Short Answer

Expert verified
The integral evaluates to \( \frac{1}{11} + \frac{9}{\ln 10} \).

Step by step solution

01

Break Down the Integral

The integral provided is \( \int_{0}^{1}(x^{10} + 10^{x}) \, dx \). It represents the sum of the integrals of \( x^{10} \) and \( 10^{x} \) over the interval \([0,1]\). Thus, we can express it as two separate integrals: \( \int_{0}^{1} x^{10} \, dx + \int_{0}^{1} 10^{x} \, dx \).
02

Integrate \( x^{10} \)

To find \( \int_{0}^{1} x^{10} \, dx \), we use the power rule for integration which states \( \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \). Applying this, we have \( \int x^{10} \, dx = \frac{x^{11}}{11} \). Evaluating this from 0 to 1 gives \( \left[ \frac{x^{11}}{11} \right]_{0}^{1} = \frac{1^{11}}{11} - \frac{0^{11}}{11} = \frac{1}{11} \).
03

Integrate \( 10^{x} \)

The integral \( \int 10^{x} \, dx \) involves an exponential function. The antiderivative of \( a^{x} \) is \( \frac{a^{x}}{\ln a} + C \). Here, \( a = 10 \), so \( \int 10^{x} \, dx = \frac{10^{x}}{\ln 10} \). Evaluating from 0 to 1 gives \( \left[ \frac{10^{x}}{\ln 10} \right]_{0}^{1} = \frac{10^{1}}{\ln 10} - \frac{10^{0}}{\ln 10} = \frac{10}{\ln 10} - \frac{1}{\ln 10} = \frac{9}{\ln 10} \).
04

Combine Results

Now, combine the results from Steps 2 and 3. The total integral is \( \frac{1}{11} + \frac{9}{\ln 10} \). This expression represents the evaluated integral over the interval \([0, 1]\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule for Integration
The power rule for integration is a fundamental tool in calculus that helps find antiderivatives of polynomial functions. It simplifies the process of integrating terms where you have a variable raised to a power. The rule states: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] where:
  • \( n \) is a constant that represents the power of \( x \).
  • \( C \) is the constant of integration, appearing because the antiderivative is not unique.
In our exercise, to integrate \( x^{10} \), we apply the power rule by setting \( n = 10 \). This gives us:
  • \( \int x^{10} \, dx = \frac{x^{11}}{11} + C \)
When we compute definite integrals, the constant \( C \) goes away as it cancels out in the evaluation process. You evaluate the antiderivative at both the upper and lower limits and subtract. For example:
  • \[ \left[ \frac{x^{11}}{11} \right]_{0}^{1} = \frac{1^{11}}{11} - \frac{0^{11}}{11} = \frac{1}{11} \]
Exponential Function Integration
Integrating exponential functions is another essential rule in calculus. Unlike polynomial functions, exponential functions often involve a base other than the natural base \( e \). The general formula for integrating an exponential function \( a^x \) is:\[ \int a^x \, dx = \frac{a^x}{\ln a} + C \]This formula derives from the fact that the derivative of \( a^x \) is \( a^x \ln a \). Key points include:
  • The base \( a \) must be a positive constant different from 1.
  • \( \ln a \) represents the natural logarithm of \( a \).
  • The constant \( C \) serves as the constant of integration for indefinite integrals.
In our exercise, we need to compute \( \int 10^x \, dx \) using this rule:
  • Set \( a = 10 \), giving \( \int 10^x \, dx = \frac{10^x}{\ln 10} + C \).
During the evaluation of the definite integral from 0 to 1, we get:
  • \[ \left[ \frac{10^x}{\ln 10} \right]_{0}^{1} = \frac{10}{\ln 10} - \frac{1}{\ln 10} = \frac{9}{\ln 10} \]
Evaluating Integrals
The process of evaluating definite integrals involves calculating the net area under a curve over a specified interval. Here鈥檚 how to approach it practically:
  • Identify that the definite integral \( \int_{a}^{b} f(x) \, dx \) calculates the area from \( x = a \) to \( x = b \).
  • First, find the antiderivative of the function.
  • Then substitute the upper limit \( b \) into the antiderivative.
  • Subtract the result of substituting the lower limit \( a \).
In our specific problem, the integral is expressed as two separate parts: \( \int_{0}^{1} x^{10} \, dx \) and \( \int_{0}^{1} 10^x \, dx \). After finding each integral result, simply add them:
  • \( \int_{0}^{1} (x^{10} + 10^x) \, dx = \frac{1}{11} + \frac{9}{\ln 10} \)
This sum evaluates the total area represented by the original integral over the interval from \( 0 \) to \( 1 \). Thus, understanding how to evaluate integrals equips you with a tool for solving a wide range of problems in calculus.

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