Find the Antiderivative f(x)=7sin(x)
The problem asks for the calculation of an antiderivative (also known as the indefinite integral) of a given function f(x). The function in this case is 7sin(x), which is a trigonometric function involving the sine of x multiplied by the constant 7. The question requires one to determine a function F(x) whose derivative with respect to x is 7sin(x).
$f \left(\right. x \left.\right) = 7 sin \left(\right. x \left.\right)$
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.
\[ F(x) = \int 7 \sin(x) \, dx \]
Step 3:
Extract the constant factor $7$ from the integral, as it does not depend on $x$.
\[ 7 \int \sin(x) \, dx \]
Step 4:
Compute the integral of $\sin(x)$ with respect to $x$, which is $-\cos(x)$.
\[ 7 \left( -\cos(x) + C \right) \]
Step 5:
Proceed to simplify the expression.
Step 5.1:
Begin simplification.
\[ 7 \left( -\cos(x) \right) + C \]
Step 5.2:
Distribute the $7$ across the $-1$.
\[ -7 \cos(x) + C \]
Step 6:
Conclude with the antiderivative of the function $f(x) = 7 \sin(x)$.
\[ F(x) = -7 \cos(x) + C \]
Knowledge Notes:
To solve for the antiderivative (also known as the indefinite integral) of a function, one must be familiar with basic integration rules and the antiderivatives of common functions. Here are the relevant knowledge points for this problem:
Constant Multiple Rule: If $k$ is a constant and $f(x)$ is a function, then the integral of $k \cdot f(x)$ is $k$ times the integral of $f(x)$.
\[ \int k f(x) \, dx = k \int f(x) \, dx \]
Antiderivative of Sine: The antiderivative of $\sin(x)$ is $-\cos(x)$, since the derivative of $\cos(x)$ is $-\sin(x)$.
\[ \int \sin(x) \, dx = -\cos(x) + C \] where $C$ is the constant of integration.
Simplification: After integrating, it's important to simplify the expression to its most basic form.
Constant of Integration: When finding an indefinite integral, one must add a constant of integration $C$, since the derivative of a constant is zero and thus the original function could have had any constant value added to it.
Integration Notation: The integral sign $\int$ is followed by the function to be integrated and then $dx$, which indicates the variable of integration.
By applying these principles, one can find the antiderivative of $7\sin(x)$ as shown in the solution steps.
