Find the Antiderivative f(x)=e^(4x)

Solution:

Step 1:

Identify the antiderivative $F(x)$ by integrating the given function $f(x)$.
$F(x) = \int f(x) \, dx$

Step 2:

Begin the integration process.
$F(x) = \int e^{4x} \, dx$

Step 3:

Substitute $u = 4x$. Consequently, $du = 4 \, dx$ or $\frac{1}{4} \, du = dx$. Use $u$ and $du$ to restate the integral.

Step 3.1:

Assign $u = 4x$ and calculate $\frac{du}{dx}$.

Step 3.1.1:

Apply differentiation to $4x$.
$\frac{d}{dx}(4x)$

Step 3.1.2:

Since $4$ is a constant, its derivative with respect to $x$ is $4 \cdot \frac{d}{dx}(x)$.
$4 \cdot \frac{d}{dx}(x)$

Step 3.1.3:

Utilize the Power Rule, which states that $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n = 1$.
$4 \cdot 1$

Step 3.1.4:

Multiply $4$ by $1$.
$4$

Step 3.2:

Reformulate the integral using $u$ and $du$.
$\int e^u \cdot \frac{1}{4} \, du$

Step 4:

Integrate the function $e^u$ multiplied by $\frac{1}{4}$.
$\int \frac{e^u}{4} \, du$

Step 5:

Extract the constant $\frac{1}{4}$ from the integral.
$\frac{1}{4} \int e^u \, du$

Step 6:

Integrate $e^u$ with respect to $u$.
$\frac{1}{4}(e^u + C)$

Step 7:

Express the result in its simplest form.
$\frac{1}{4} e^u + C$

Step 8:

Substitute back $u = 4x$ into the antiderivative.
$\frac{1}{4} e^{4x} + C$

Step 9:

Conclude with the antiderivative of $f(x) = e^{4x}$.
$F(x) = \frac{1}{4} e^{4x} + C$

Knowledge Notes:

The process of finding the antiderivative, also known as the indefinite integral, involves reversing the differentiation process. The antiderivative of a function $f(x)$ is a function $F(x)$ whose derivative is $f(x)$, denoted as $F(x) = \int f(x) \, dx$. Here are some relevant knowledge points:

1. Exponential Function: The exponential function $e^x$ has a unique property where its derivative is itself, $\frac{d}{dx}e^x = e^x$.

2. Substitution Rule: This rule is used to simplify integration by substituting a part of the integrand with a new variable. If $u = g(x)$, then $du = g'(x)dx$.

3. Power Rule for Differentiation: This rule states that the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$. It is used to find the derivative of $u$ with respect to $x$ when $u$ is a power function of $x$.

4. Constant Multiple Rule: This rule allows a constant to be factored out of the integral, which simplifies the integration process. If $k$ is a constant, then $\int k \cdot f(x) \, dx = k \int f(x) \, dx$.

5. Integration of Exponential Functions: The integral of $e^u$ with respect to $u$ is $e^u + C$, where $C$ is the constant of integration.

6. Back-Substitution: After integrating with respect to the substitution variable $u$, we replace $u$ with the original expression in terms of $x$ to obtain the antiderivative in terms of the original variable.