Find the Antiderivative f(x)=9e^x+8sec(x)^2

Solution:

Step 1:

Identify the antiderivative $F(x)$ by integrating the given function $f(x)$.

$F(x) = \int f(x) dx$

Step 2:

Write down the integral that needs to be solved.

$F(x) = \int (9e^x + 8\sec^2(x)) dx$

Step 3:

Decompose the integral into two separate integrals.

$\int 9e^x dx + \int 8\sec^2(x) dx$

Step 4:

Extract the constant factor $9$ from the first integral.

$9\int e^x dx + \int 8\sec^2(x) dx$

Step 5:

Integrate $e^x$ with respect to $x$ to get $e^x$.

$9(e^x + C) + \int 8\sec^2(x) dx$

Step 6:

Extract the constant factor $8$ from the second integral.

$9(e^x + C) + 8\int \sec^2(x) dx$

Step 7:

Recognize that the integral of $\sec^2(x)$ is $\tan(x)$.

$9(e^x + C) + 8(\tan(x) + C)$

Step 8:

Combine the terms to simplify.

$9e^x + 8\tan(x) + C$

Step 9:

Present the final antiderivative of the function $f(x) = 9e^x + 8\sec^2(x)$.

$F(x) = 9e^x + 8\tan(x) + C$

Knowledge Notes:

To solve for the antiderivative of a function $f(x)$, we follow these steps:

1. Understanding Antiderivatives: An antiderivative of a function $f(x)$ is another function $F(x)$ such that $F'(x) = f(x)$. The process of finding $F(x)$ is called integration.

2. Constants and Integration: When integrating a function multiplied by a constant, the constant can be factored out of the integral. For example, $\int cf(x)dx = c\int f(x)dx$ where $c$ is a constant.

3. Basic Integration Rules: Some basic functions have well-known antiderivatives, such as:

   • $\int e^x dx = e^x + C$
   • $\int \sec^2(x) dx = \tan(x) + C$ where $C$ represents the constant of integration.

4. Linearity of Integration: The integral of a sum of functions is equal to the sum of their integrals. For example, $\int (f(x) + g(x))dx = \int f(x)dx + \int g(x)dx$.

5. Indefinite Integrals: The antiderivative is called an indefinite integral because it includes an arbitrary constant $C$. This constant represents the family of all antiderivatives of the function.

6. Trigonometric Integrals: Knowing the derivatives of trigonometric functions helps in finding their antiderivatives. For instance, since the derivative of $\tan(x)$ is $\sec^2(x)$, the antiderivative of $\sec^2(x)$ is $\tan(x) + C$.

By applying these principles, we can find the antiderivative of the given function $f(x) = 9e^x + 8\sec^2(x)$.