Find the Antiderivative g(t)=e^(-7t)

Solution:

Step 1:

Identify the antiderivative $G(t)$ by integrating the given function $g(t)$.

$$G(t) = \int g(t) \, dt$$

Step 2:

Write down the integral that needs to be solved.

$$G(t) = \int e^{-7t} \, dt$$

Step 3:

Perform a substitution to simplify the integral. Let $u = -7t$.

Step 3.1:

Calculate the derivative of $u$ with respect to $t$.

Step 3.1.1:

Take the derivative of $-7t$.

$$\frac{du}{dt} = \frac{d}{dt}(-7t)$$

Step 3.1.2:

Since $-7$ is a constant, its derivative is $-7$ times the derivative of $t$.

$$-7 \frac{d}{dt}(t)$$

Step 3.1.3:

Apply the Power Rule for differentiation, which states that the derivative of $t^n$ is $nt^{n-1}$, where $n=1$.

$$-7 \cdot 1$$

Step 3.1.4:

Multiply $-7$ by $1$.

$$-7$$

Step 3.2:

Express the integral in terms of $u$ and $du$.

$$\int e^u \cdot \frac{1}{-7} \, du$$

Step 4:

Simplify the integral.

Step 4.1:

Extract the negative sign from the fraction.

$$\int e^u \left(-\frac{1}{7}\right) \, du$$

Step 4.2:

Combine the exponential function $e^u$ with the constant $\frac{1}{7}$.

$$\int -\frac{e^u}{7} \, du$$

Step 5:

Since $-1$ is a constant, it can be taken out of the integral.

$$-\int \frac{e^u}{7} \, du$$

Step 6:

Extract the constant $\frac{1}{7}$ from the integral.

$$-\left(\frac{1}{7} \int e^u \, du\right)$$

Step 7:

Integrate $e^u$ with respect to $u$.

$$-\frac{1}{7}(e^u + C)$$

Step 8:

Simplify the expression.

$$-\frac{1}{7}e^u + C$$

Step 9:

Substitute back the original variable $t$ for $u$.

$$-\frac{1}{7}e^{-7t} + C$$

Step 10:

Conclude with the antiderivative of the function $g(t) = e^{-7t}$.

$$G(t) = -\frac{1}{7}e^{-7t} + C$$

Knowledge Notes:

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

1. Integration by Substitution: This technique, also known as u-substitution, is used to simplify integrals by changing the variable of integration. It is the reverse process of the Chain Rule in differentiation.

2. Power Rule for Integration: The Power Rule states that the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1}$, provided that $n \neq -1$.

3. Exponential Function Integration: The integral of $e^x$ with respect to $x$ is $e^x + C$, where $C$ is the constant of integration.

4. Constant Multiple Rule: A constant multiplier can be pulled out of the integral. If $k$ is a constant and $f(x)$ is a function, then $\int k \cdot f(x) \, dx = k \cdot \int f(x) \, dx$.

5. Adding the Constant of Integration: When finding an indefinite integral, it is important to add a constant term $C$ at the end of the integration process, as there are infinitely many antiderivatives differing by a constant.

6. LaTeX for Mathematical Notation: In the solution, LaTeX is used to format mathematical expressions, which allows for clear and precise rendering of formulas and equations.