Find the Antiderivative f(x)=sin(9x)

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.

$$F(x)=\int \sin (9x)dx$$

Step 3:

Perform a substitution to simplify the integral. Set $u=9x$.

Step 3.1:

Determine $\frac{du}{dx}$ by differentiating $u$.

Step 3.1.1:

Differentiate $9x$.

$$\frac{d}{dx}(9x)$$

Step 3.1.2:

Apply the constant multiple rule, differentiating $9x$ with respect to $x$.

$$9\frac{d}{dx}(x)$$

Step 3.1.3:

Use the Power Rule, which states $\frac{d}{dx}(x^n)=n{x}^{n-1}$, where $n=1$.

$$9\cdot 1$$

Step 3.1.4:

Calculate the product of $9$ and $1$.

$$9$$

Step 3.2:

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

$$\int \sin (u)\frac{1}{9}du$$

Step 4:

Combine $\sin (u)$ with the constant $\frac{1}{9}$.

$$\int \frac{\sin (u)}{9}du$$

Step 5:

Extract the constant $\frac{1}{9}$ from the integral as it is constant with respect to $u$.

$$\frac{1}{9}\int \sin (u)du$$

Step 6:

Integrate $\sin (u)$ with respect to $u$ to find the antiderivative.

$$\frac{1}{9}(-\cos (u)+C)$$

Step 7:

Simplify the expression.

Step 7.1:

Simplify the antiderivative.

$$\frac{1}{9}(-\cos (u))+C$$

Step 7.2:

Combine the constant $\frac{1}{9}$ with $\cos (u)$.

$$-\frac{\cos (u)}{9}+C$$

Step 8:

Substitute back $u=9x$ into the antiderivative.

$$-\frac{\cos (9x)}{9}+C$$

Step 9:

Rearrange the terms for the final antiderivative.

$$-\frac{1}{9}\cos (9x)+C$$

Step 10:

Conclude with the antiderivative of the function $f(x)=\sin (9x)$.

$$F(x)=-\frac{1}{9}\cos (9x)+C$$

Knowledge Notes:

The process of finding the antiderivative involves several key knowledge points:

1. Indefinite Integral: The antiderivative of a function $f(x)$ is found by integrating the function, denoted as $\int f(x)dx$. The result is a family of functions $F(x)$ plus an arbitrary constant $C$.

2. Substitution Rule: This is a method used to simplify integrals by substituting a part of the integral with a new variable. This often makes the integral easier to solve.

3. Differentiation Rules:

   • Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

   • Power Rule: The derivative of $x^n$ is $n{x}^{n-1}$. This rule is used when differentiating polynomials.

4. Integration of Trigonometric Functions: The integral of $\sin (x)$ is $-\cos (x)$, and this rule is applied when integrating functions involving the sine function.

5. Simplification: After integrating and applying substitution, the expression is often simplified by combining like terms and substituting back the original variables.

By understanding and applying these concepts, one can solve a wide range of antiderivative problems in calculus.