Integrate Using u-Substitution integral from 0 to 2 of e^(2x) with respect to x

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 $\frac{d}{dx}[x^n]=n{x}^{n-1}$ 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 $u$: $\frac{1}{2}{e}^u{|}_0^4$.

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:

$26.79907501\dots$

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 $n{x}^{n-1}$.

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.