Find the Antiderivative f(x)=e^(12x)
The question is asking for the antiderivative (also known as the indefinite integral) of the function f(x) = e^(12x). In calculus, finding an antiderivative means determining a function F(x) such that its derivative with respect to x is equal to the original function f(x). In this case, you need to provide the function F(x) that, when differentiated, results in f(x) = e^(12x).
$f \left(\right. x \left.\right) = e^{12 x}$
Answer
Solution:
Step 1:
Identify the antiderivative $F(x)$ by integrating the function $f(x)$.
$$F(x) = \int f(x) \, dx$$
Step 2:
Begin by setting up the integral.
$$F(x) = \int e^{12x} \, dx$$
Step 3:
Introduce a substitution with $u = 12x$. Consequently, $du = 12 \, dx$ or $dx = \frac{1}{12} \, du$.
Step 3.1:
Define the substitution $u = 12x$ and compute $\frac{du}{dx}$.
Step 3.1.1:
Take the derivative of $12x$.
$$\frac{d}{dx}(12x)$$
Step 3.1.2:
Apply the constant multiple rule, since 12 is a constant.
$$12 \frac{d}{dx}(x)$$
Step 3.1.3:
Use the Power Rule for differentiation, which states that $\frac{d}{dx}(x^n) = nx^{n-1}$ where $n = 1$.
$$12 \cdot 1$$
Step 3.1.4:
Calculate the product of 12 and 1.
$$12$$
Step 3.2:
Reformulate the integral using $u$ and $du$.
$$\int e^u \frac{1}{12} \, du$$
Step 4:
Combine the exponential function with the constant.
$$\int \frac{e^u}{12} \, du$$
Step 5:
Since $\frac{1}{12}$ is a constant, it can be factored out of the integral.
$$\frac{1}{12} \int e^u \, du$$
Step 6:
Integrate $e^u$ with respect to $u$.
$$\frac{1}{12}(e^u + C)$$
Step 7:
Simplify the expression.
$$\frac{1}{12} e^u + C$$
Step 8:
Substitute back the original variable, replacing $u$ with $12x$.
$$\frac{1}{12} e^{12x} + C$$
Step 9:
Conclude with the antiderivative of $f(x) = e^{12x}$.
$$F(x) = \frac{1}{12} e^{12x} + C$$
Knowledge Notes:
The process of finding the antiderivative, also known as the indefinite integral, involves reversing the process of differentiation. Here are the relevant knowledge points used in the solution:
Indefinite Integral: The antiderivative or indefinite integral of a function $f(x)$ is denoted by $\int f(x) \, dx$ and represents the family of all functions whose derivative is $f(x)$.
Exponential Function: The exponential function $e^x$ has the unique property that its derivative is itself, i.e., $\frac{d}{dx}e^x = e^x$.
Substitution Rule: This is a technique for evaluating integrals. If a function $u=g(x)$ is substituted into an integral, the differential $dx$ is replaced by $du$, where $du = g'(x) \, dx$.
Constant Multiple Rule: When taking the derivative or integral of a constant multiplied by a function, the constant can be pulled out of the differentiation or integration.
Power Rule for Integration: The power rule for integration is the reverse of the power rule for differentiation. For an exponential function $e^u$, the integral is $e^u + C$, where $C$ is the constant of integration.
Constant of Integration: When finding an indefinite integral, a constant $C$ is added to represent the family of all antiderivatives, since the derivative of a constant is zero.
In the given solution, the substitution method is used to simplify the integration of the exponential function $e^{12x}$, and the constant multiple rule allows the constant $\frac{1}{12}$ to be factored out of the integral. The final answer represents the antiderivative of the given function $f(x) = e^{12x}$.
