Find the Antiderivative f(x)=6x+sin(x)
The problem is asking for the calculation of the antiderivative (or indefinite integral) of a given function, which in this case is \( f(x) = 6x + \sin(x) \). This means you are being asked to find a function \( F(x) \) whose derivative with respect to \( x \) is \( 6x + \sin(x) \). The antiderivative is a reverse operation of taking a derivative, and in this case, it involves integrating a linear term \( 6x \) and a trigonometric function \( \sin(x) \).
$f \left(\right. x \left.\right) = 6 x + sin \left(\right. x \left.\right)$
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.
$$F(x) = \int (6x + \sin(x)) \, dx$$
Step 3:
Decompose the integral into simpler parts.
$$\int 6x \, dx + \int \sin(x) \, dx$$
Step 4:
Extract the constant $6$ from the integral as it is not dependent on $x$.
$$6 \int x \, dx + \int \sin(x) \, dx$$
Step 5:
Apply the Power Rule of integration to find the integral of $x$.
$$6 \left( \frac{x^2}{2} + C \right) + \int \sin(x) \, dx$$
Step 6:
Integrate $\sin(x)$ to get $-\cos(x)$.
$$6 \left( \frac{x^2}{2} + C \right) - \cos(x) + C$$
Step 7:
Simplify the expression.
Step 7.1:
Combine the constant with $x^2$.
$$6 \left( \frac{x^2}{2} + C \right) - \cos(x) + C$$
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:
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.
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.
Constant Multiple Rule: If $k$ is a constant and $f(x)$ is a function, then $\int k f(x) \, dx = k \int f(x) \, dx$. This is why we can pull the constant $6$ out of the integral.
Power Rule for Integration: For any real number $n \neq -1$, the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1} + C$. In our case, integrating $x$ gives us $\frac{x^2}{2} + C$.
Integral of Trigonometric Functions: The integral of $\sin(x)$ is $-\cos(x) + C$. This is a standard result from the list of basic integrals.
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.
