Find the Antiderivative g(x)=sin(x)+14cos(x)
The given problem is asking for the calculation of an antiderivative, also known as the indefinite integral, of the function g(x) = sin(x) + 14cos(x). The question requires one to find a function F(x) such that its derivative with respect to x is equal to g(x), meaning that F'(x) = sin(x) + 14cos(x). The solution process would typically involve using the integral calculus methods to determine this antiderivative.
$g \left(\right. x \left.\right) = sin \left(\right. x \left.\right) + 14 cos \left(\right. x \left.\right)$
Answer
Solution:
Step 1:
Identify the antiderivative $G(x)$ by integrating the function $g(x)$.
$$G(x) = \int g(x) \, dx$$
Step 2:
Write down the integral that needs to be solved.
$$G(x) = \int (\sin(x) + 14\cos(x)) \, dx$$
Step 3:
Decompose the integral into a sum of integrals.
$$\int \sin(x) \, dx + \int 14\cos(x) \, dx$$
Step 4:
Integrate $\sin(x)$ with respect to $x$ to get $-\cos(x)$.
$$-\cos(x) + C + \int 14\cos(x) \, dx$$
Step 5:
Factor out the constant $14$ from the integral.
$$-\cos(x) + C + 14\int \cos(x) \, dx$$
Step 6:
Integrate $\cos(x)$ with respect to $x$ to obtain $\sin(x)$.
$$-\cos(x) + C + 14(\sin(x) + C)$$
Step 7:
Combine the terms to simplify the expression.
$$-\cos(x) + 14\sin(x) + C$$
Step 8:
Present the final antiderivative of $g(x) = \sin(x) + 14\cos(x)$.
$$G(x) = -\cos(x) + 14\sin(x) + C$$
Knowledge Notes:
To solve for the antiderivative of a function, we follow these steps:
Understanding Antiderivatives: An antiderivative of a function $f(x)$ is a function $F(x)$ whose derivative is $f(x)$. In other words, $F'(x) = f(x)$. We often denote antiderivatives with a capital letter and the original function with a lowercase letter.
Setting Up the Integral: The process of finding an antiderivative is equivalent to evaluating an indefinite integral. We set up the integral by placing the function inside the integral sign.
Linearity of Integration: The integral of a sum of functions is equal to the sum of the integrals of each function. This property allows us to split the integral of a sum into separate integrals.
Integrating Trigonometric Functions: The antiderivatives of basic trigonometric functions are well-known. For example, the antiderivative of $\sin(x)$ is $-\cos(x)$, and the antiderivative of $\cos(x)$ is $\sin(x)$.
Constants of Integration: When we integrate a function, we add a constant of integration $C$ because the derivative of a constant is zero. This constant represents the family of all antiderivatives of the function.
Constant Factor Rule: If a constant is multiplied by a function, it can be factored out of the integral. This is known as the constant factor rule and is a consequence of the linearity of integration.
Simplifying the Expression: After integrating, we combine like terms and constants to simplify the expression for the antiderivative.
Final Antiderivative: The final step is to write down the simplified antiderivative, which includes the constant of integration.
