Find the Antiderivative f(x)=3sin(x)

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 $\sin (x)$.

$$3(-\cos (x)+C)$$

Step 5:

Finalize the expression.

Step 5.1:

Apply the constant multiplier.

$$3(-\cos (x))+C$$

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:

1. Antiderivative: The antiderivative of a function $f(x)$ is another function $F(x)$ such that $F^{\prime }(x)=f(x)$. It is represented by the integral symbol with no bounds, indicating an indefinite integral.

2. 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:

   $$\int k\cdot g(x)dx=k\int g(x)dx$$

3. 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 $\sin (x)$.

4. 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.

5. 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.