Find the Antiderivative f(x)=6x+sin(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.

$$F(x)=\int (6x+\sin (x))dx$$

Step 3:

Decompose the integral into simpler parts.

$$\int 6xdx+\int \sin (x)dx$$

Step 4:

Extract the constant $6$ from the integral as it is not dependent on $x$.

$$6\int xdx+\int \sin (x)dx$$

Step 5:

Apply the Power Rule of integration to find the integral of $x$.

$$6(\frac{x^2}{2}+C)+\int \sin (x)dx$$

Step 6:

Integrate $\sin (x)$ to get $-\cos (x)$.

$$6(\frac{x^2}{2}+C)-\cos (x)+C$$

Step 7:

Simplify the expression.

Step 7.1:

Combine the constant with $x^2$.

$$6(\frac{x^2}{2}+C)-\cos (x)+C$$

Step 7.2:

Final simplification.

$$3x^2-\cos (x)+C$$

Step 8:

Conclude with the antiderivative of $f(x)=6x+\sin (x)$.

$$F(x)=3x^2-\cos (x)+C$$

Knowledge Notes:

To solve for the antiderivative of a function, you need to be familiar with the following concepts:

1. Indefinite Integral: The antiderivative of a function $f(x)$ is represented by the indefinite integral $\int f(x)dx$. It includes an arbitrary constant $C$ since the derivative of a constant is zero.

2. Linearity of Integration: The integral of a sum of functions is equal to the sum of their integrals. This allows us to split the integral of $6x+\sin (x)$ into separate integrals.

3. Constant Multiple Rule: If $k$ is a constant and $f(x)$ is a function, then $\int kf(x)dx=k\int f(x)dx$. This is why we can pull the constant $6$ out of the integral.

4. Power Rule for Integration: For any real number $n\ne -1$, the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1}+C$. In our case, integrating $x$ gives us $\frac{x^2}{2}+C$.

5. Integral of Trigonometric Functions: The integral of $\sin (x)$ is $-\cos (x)+C$. This is a standard result from the list of basic integrals.

6. Simplification: After integrating, we combine like terms and simplify the expression to get the final antiderivative.

Understanding these principles is essential for solving integration problems in calculus.