Integrate Using u-Substitution integral from 0 to 2 of e^(2x) with respect to x
The question is asking how to compute the integral of the exponential function e raised to the power of (2x) from the lower limit of 0 to the upper limit of 2, using a method of integration called "u-substitution". This technique involves substituting a part of the integrand (the expression inside the integral) with a new variable u, and then changing the differential accordingly, to simplify the integration process. The problem requires you to identify the appropriate substitution, change the limits of integration if necessary, perform the integration with respect to the new variable u, and then substitute back to the original variable x to get the final result.
$\int_{0}^{2} e^{2 x} d x$
Answer
Solution:
Step 1: Choose a substitution variable
Assign $u = 2x$. Compute the differential $du$ and express $dx$ in terms of $du$.
Step 1.1: Define the substitution
Set $u = 2x$ and calculate $\frac{du}{dx}$.
Step 1.1.1: Differentiate the expression
Take the derivative of $2x$ with respect to $x$: $\frac{d}{dx}[2x]$.
Step 1.1.2: Apply the constant rule in differentiation
Since $2$ is a constant, its derivative is $2\frac{d}{dx}[x]$.
Step 1.1.3: Use the Power Rule for differentiation
Apply the Power Rule, which states $\frac{d}{dx}[x^n] = nx^{n-1}$ where $n=1$: $2 \cdot 1$.
Step 1.1.4: Simplify the derivative
Multiply $2$ by $1$: $2$.
Step 1.2: Substitute the lower limit into $u$
Replace $x$ with the lower limit in $u = 2x$: $u_{\text{lower}} = 2 \cdot 0$.
Step 1.3: Calculate the new lower limit
Compute $2$ times $0$: $u_{\text{lower}} = 0$.
Step 1.4: Substitute the upper limit into $u$
Replace $x$ with the upper limit in $u = 2x$: $u_{\text{upper}} = 2 \cdot 2$.
Step 1.5: Calculate the new upper limit
Compute $2$ times $2$: $u_{\text{upper}} = 4$.
Step 1.6: Prepare to use the new limits for integration
Use the values of $u_{\text{lower}}$ and $u_{\text{upper}}$ to evaluate the definite integral: $u_{\text{lower}} = 0$, $u_{\text{upper}} = 4$.
Step 1.7: Rewrite the integral with the new variable and limits
Express the integral in terms of $u$, $du$, and the new integration limits: $\int_{0}^{4} e^u \frac{1}{2} du$.
Step 2: Combine the integrand with the constant
Integrate $\int_{0}^{4} \frac{e^u}{2} du$.
Step 3: Factor out the constant from the integral
Extract $\frac{1}{2}$ from the integral: $\frac{1}{2} \int_{0}^{4} e^u du$.
Step 4: Integrate the exponential function
Find the antiderivative of $e^u$ with respect to $u$: $\frac{1}{2} e^u \bigg|_{0}^{4}$.
Step 5: Evaluate the definite integral
Simplify the resulting expression.
Step 5.1: Apply the Fundamental Theorem of Calculus
Evaluate $e^u$ at the limits $4$ and $0$: $\frac{1}{2} (e^4 - e^0)$.
Step 5.2: Simplify the expression
Recognize that any number raised to $0$ equals $1$: $\frac{1}{2} (e^4 - 1)$.
Step 5.3: Finalize the simplification
Multiply $-1$ by $1$: $\frac{1}{2} (e^4 - 1)$.
Step 6: Present the result in different forms
Exact Form:
$\frac{1}{2} \cdot (e^4 - 1)$
Decimal Form:
$26.79907501 \ldots$
Knowledge Notes:
The process of u-substitution is a technique used in calculus to simplify the integration of composite functions. It involves choosing a new variable, $u$, that is a function of $x$, and then expressing $dx$ in terms of $du$. This allows for the rewriting of the integral in terms of $u$, which can often be easier to integrate.
The Power Rule for differentiation is a basic rule in calculus that states that the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.
The Fundamental Theorem of Calculus links the concept of differentiation with that of integration and allows us to evaluate definite integrals by finding the antiderivative of the function and then using the upper and lower limits of integration.
In this problem, we used u-substitution to transform the integral of $e^{2x}$ into a simpler form that could be easily integrated. After finding the antiderivative, we evaluated it at the new limits of integration to find the exact value of the original integral.
