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Magazine Chapter 11: Problem 17
Evaluate the definite integral. $$\int_{0}^{1}\left(e^{2 x}+x^{2}+1\right) d x$$
Short Answer
Expert verified
The value of the definite integral \(\int_{0}^{1}\left(e^{2 x}+x^{2}+1\right) d x\) is \(\frac{1}{2} e^2 + \frac{1}{3}\).
Step by step solution
01
Identify the antiderivative of each part of the function
We have three parts in our integral function: \(e^{2x}\), \(x^2\), and 1. First, let's find the antiderivatives for each of these parts.
1. Antiderivative of \(e^{2x}\): Using the rule for exponential functions, the antiderivative of \(e^{2x}\) is \(\frac{1}{2} e^{2x}\).
2. Antiderivative of \(x^2\): Using the power rule, the antiderivative of \(x^2\) is \(\frac{1}{3} x^3\).
3. Antiderivative of 1: The antiderivative of a constant is just the constant multiplied by x. So, the antiderivative of 1 is x.
Now that we have the antiderivatives of each part, we can write the antiderivative of the entire function.
02
Construct the antiderivative of the entire function
Combine the antiderivatives of each part to form the antiderivative of the whole function F(x):
F(x) = \(\frac{1}{2} e^{2x} + \frac{1}{3} x^3 + x\)
03
Evaluate the antiderivative at the limits
Now, we need to evaluate F(x) at the given limits (0 and 1) and subtract the results. This will give us the value of the definite integral.
F(1) = \(\frac{1}{2} e^{2(1)} + \frac{1}{3} (1)^3 + (1) = \frac{1}{2} e^2 + \frac{1}{3} + 1\)
F(0) = \(\frac{1}{2} e^{2(0)} + \frac{1}{3} (0)^3 + (0) = \frac{1}{2} e^0 = \frac{1}{2}\)
Finally, subtract F(0) from F(1) to find the value of the integral:
\(\int_{0}^{1}\left(e^{2 x}+x^{2}+1\right) d x = F(1) - F(0) = \left(\frac{1}{2} e^2 + \frac{1}{3} + 1\right) - \frac{1}{2} = \frac{1}{2} e^2 + \frac{1}{3}\)
The value of the definite integral is \(\frac{1}{2} e^2 + \frac{1}{3}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Antiderivative
An antiderivative, sometimes called an indefinite integral, is a function whose derivative gives back the original function. Finding an antiderivative is like doing the reverse of differentiation. In the context of definite integrals, as seen in our exercise, we find the antiderivative to then evaluate it over a specific interval.
For instance:
For instance:
- For the function \( e^{2x} \), the antiderivative is \( \frac{1}{2} e^{2x} \). Here, a multiplying factor comes into play because of the chain rule in reverse.
- For \( x^2 \), the antiderivative is achieved using the power rule which results in \( \frac{1}{3} x^3 \).
- Lastly, for the constant function, 1, the antiderivative is simply \( x \) since the derivative of \( x \) is 1.
Exponential Function
Exponential functions are mathematical functions of the form \( e^{kx} \), where \( e \) is Euler's number (approximately 2.71828) and \( k \) is a constant.
These functions have unique properties. Their rate of change is directly proportional to their value, making them critical in fields like growth phenomena and decay processes.
When integrating exponential functions like \( e^{2x} \), it's essential to adjust for the constant \( k \) in the exponent:
These functions have unique properties. Their rate of change is directly proportional to their value, making them critical in fields like growth phenomena and decay processes.
When integrating exponential functions like \( e^{2x} \), it's essential to adjust for the constant \( k \) in the exponent:
- The antiderivative of \( e^{2x} \) is \( \frac{1}{2} e^{2x} \). The \( \frac{1}{2} \) comes from dividing by the constant \( 2 \) (from the exponent), which is derived from the chain rule reversed.
Power Rule
The power rule for integration is a straightforward technique, particularly useful when dealing with polynomial functions.
It states that for any power of \( x \) (except \( x^{-1} \)), you increase the exponent by 1 and then divide by the new exponent.
It states that for any power of \( x \) (except \( x^{-1} \)), you increase the exponent by 1 and then divide by the new exponent.
- For example, given \( x^n \), the antiderivative is \( \frac{1}{n+1} x^{n+1} \).
- In our exercise, we used the power rule to find the antiderivative of \( x^2 \) to be \( \frac{1}{3} x^3 \), since \( n = 2 \).
Integration Limits
Integration limits are the bounds within which we evaluate the integral. When dealing with definite integrals, as in our task, these limits give the interval where we need to calculate the antiderivative.
- In our example, the integration limits are 0 and 1, meaning we evaluate the antiderivative at both points.
- The process follows these steps: Calculate \( F(x) \) at the upper limit, subtract the value of \( F(x) \) at the lower limit.
