Find the Antiderivative f(x)=7sin(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 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 $-\cos (x)$.

$$7(-\cos (x)+C)$$

Step 5:

Proceed to simplify the expression.

Step 5.1:

Begin simplification.

$$7(-\cos (x))+C$$

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:

1. 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 $f(x)$.

   $$\int kf(x)dx=k\int f(x)dx$$

2. 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.

3. Simplification: After integrating, it's important to simplify the expression to its most basic form.

4. 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.

5. 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.