Find the Antiderivative f(x)=sin(9x)
The question asks for the calculation of an antiderivative (also known as an indefinite integral) of the function f(x) = sin(9x). This requires the integration of the sine function where the argument of the sine is 9 times x, implying that one would use integral calculus techniques to find a function F(x) such that F'(x) = sin(9x). The person solving this problem should look for an expression that, when differentiated, yields the original function f(x).
$f \left(\right. x \left.\right) = sin \left(\right. 9 x \left.\right)$
Answer
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 $\frac{d}{dx}(x^n) = nx^{n-1}$, 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:
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 $C$.
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.
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 $nx^{n-1}$. This rule is used when differentiating polynomials.
Integration of Trigonometric Functions: The integral of $\sin(x)$ is $-\cos(x)$, and this rule is applied when integrating functions involving the sine function.
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.
