Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 4: Problem 96
Evaluate. $$\int_{0}^{1}(x+2)^{3} d x$$
Short Answer
Expert verified
16.25
Step by step solution
01
Identify the integral form
The given integral is \(\textstyle \int_{0}^{1}(x+2)^{3} d x \). Since it is a polynomial function within the integrand, it can be integrated by applying the power rule.
02
Expand the polynomial
Before integrating, expand the integrand \((x+2)^{3}\). Using the binomial expansion formula \(a + b\)^3 = a^3 + 3a^2b + 3ab^2 + b^3, the inner term becomes: \((x+2)^{3} = x^3 + 6x^2 + 12x + 8\).
03
Integrate term by term
Now integrate each term of the expanded polynomial separately: \(\textstyle \int_{0}^{1}(x^3 + 6x^2 + 12x + 8) d x\). The integration is performed as follows: \(\textstyle \int_{0}^{1} x^3 dx + \int_{0}^{1} 6x^2 dx + \int_{0}^{1} 12x dx + \int_{0}^{1} 8 dx \).
04
Apply the power rule of integration
Integrate each term using the power rule \( \int x^n dx = \frac{x^{n+1}}{n+1} \): \(\textstyle \int x^3 dx = \frac{x^4}{4}\), \(\textstyle \int 6x^2 dx = 6 \frac{x^3}{3} = 2x^3\), \(\textstyle \int 12x dx = 12 \frac{x^2}{2} = 6x^2\), and \(\textstyle \int 8 dx = 8x\).
05
Evaluate the definite integral
Using the integrated expressions, evaluate them from 0 to 1: \(\left [ \frac{x^4}{4} + 2x^3 + 6x^2 + 8x \right ]_{0}^{1} \). Substitute x = 1 and x = 0: \( \left ( \frac{1^4}{4} + 2(1)^3 + 6(1)^2 + 8(1) \right ) - \left ( \frac{0^4}{4} + 2(0)^3 + 6(0)^2 + 8(0) \right ) \).
06
Simplify and calculate the result
Simplify the expression: \( \left ( \frac{1}{4} + 2 + 6 + 8 \right ) - \left ( 0 \right ) = \frac{1}{4} + 2 + 6 + 8 = 16.25 \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Integration
Polynomial integration is the process of finding the integral (or antiderivative) of a polynomial function. Here, a polynomial is an expression comprised of variables and coefficients, combined using addition, subtraction, and multiplication. For example, in the integral \( \textstyle \int_{0}^{1}(x+2)^{3} dx \), the expression \( (x+2)^3 \) is initially a polynomial that we need to expand before integrating.
To integrate a polynomial:
To integrate a polynomial:
- Expand the polynomial, if necessary
- Identify the individual terms in the polynomial
- Integrate each term separately using the proper rules
Power Rule
The power rule is a fundamental tool for integrating polynomial functions and states that the integral of \( x^n \), where n is a constant, is given by:
\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \]
Here, C represents the constant of integration when dealing with indefinite integrals. However, for definite integrals, we do not add C since the bounds essentially cancel out any constants.
For example, if we want to integrate \( x^3 \), using the power rule we get:
\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \]
Here, C represents the constant of integration when dealing with indefinite integrals. However, for definite integrals, we do not add C since the bounds essentially cancel out any constants.
For example, if we want to integrate \( x^3 \), using the power rule we get:
- Increase the exponent by 1, making it 4
- Divide by the new exponent: \frac{x^4}{4}\
Binomial Expansion
The binomial expansion theorem is a way of expanding expressions that are raised to a power, such as \( (a+b)^n \). This expansion is crucial when dealing with integrals of polynomial expressions raised to a power.
The binomial theorem states:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
Using this, we can expand any binomial into a polynomial. For example, in our specific exercise:
The binomial theorem states:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
Using this, we can expand any binomial into a polynomial. For example, in our specific exercise:
- \
