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 $F^{\prime }(x)=f(x)$. 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 $f(x)=9e^x+8{\sec }^2(x)$.