Find the Antiderivative f(x)=3sin(x)
The problem is asking to determine the antiderivative, or the indefinite integral, of the function f(x) = 3sin(x). This involves finding a function F(x) such that its derivative F'(x) is equal to 3sin(x). The task does not require evaluating the integral for any specific bounds but simply finding the general form of the antiderivative.
$f \left(\right. x \left.\right) = 3 sin \left(\right. x \left.\right)$
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 the integral that needs to be solved.
$$ F(x) = \int 3 \sin(x) \, dx $$
Step 3:
Extract the constant multiplier from the integral.
$$ 3 \int \sin(x) \, dx $$
Step 4:
Compute the integral of $\sin(x)$.
$$ 3 \left( -\cos(x) + C \right) $$
Step 5:
Finalize the expression.
Step 5.1:
Apply the constant multiplier.
$$ 3 \left( -\cos(x) \right) + C $$
Step 5.2:
Distribute the constant $-3$.
$$ -3 \cos(x) + C $$
Step 6:
Present the final antiderivative of $f(x) = 3 \sin(x)$.
$$ F(x) = -3 \cos(x) + C $$
Knowledge Notes:
The process of finding the antiderivative, or the indefinite integral, involves reversing the differentiation process. Here are some relevant knowledge points:
Antiderivative: The antiderivative of a function $f(x)$ is another function $F(x)$ such that $F'(x) = f(x)$. It is represented by the integral symbol with no bounds, indicating an indefinite integral.
Integral of a Constant Multiplier: When a function being integrated is multiplied by a constant, the constant can be factored out of the integral. If $k$ is a constant and $g(x)$ is a function, then:
$$ \int k \cdot g(x) \, dx = k \int g(x) \, dx $$
Integral of Sine: The integral of $\sin(x)$ with respect to $x$ is $-\cos(x)$, which is a standard result from integral tables. This is due to the fact that the derivative of $-\cos(x)$ is $\sin(x)$.
Constant of Integration: When finding an indefinite integral, we add a constant term $C$ at the end of the expression. This constant accounts for the fact that the antiderivative is not unique鈥攁ny constant added to a function does not change its derivative.
Simplification: After computing the integral, it is often necessary to simplify the expression to make it more readable or to apply it in further calculations.
In this problem, the antiderivative of $f(x) = 3 \sin(x)$ is found by integrating $\sin(x)$ and multiplying the result by the constant factor $3$. The constant of integration $C$ is added to represent the family of all antiderivatives of the function.
