Question

For the following functions \(f\), find the anti-derivative \(F\) that satisfies the given condition. $$f(x)=8 x^{3}+\sin x ; F(0)=2$$

Short answer

Question: Find the antiderivative, \(F(x)\), of the function \(f(x) = 8x^3 + \sin x\), given that \(F(0) = 2\). Answer: The antiderivative \(F(x)\) is \(F(x) = 2x^4 - \cos x + 1\).

Step-by-step solution

Find the antiderivative of \(f(x)\)

To find the antiderivative \(F(x)\) of the given function \(f(x) = 8x^3 + \sin x\), we need to find the function whose derivative is \(f(x)\). This can be done using the standard rules of integration like the power rule and integration of elementary functions: $$F(x) = \int (8x^3 + \sin x)\, dx$$ $$F(x) = \int 8x^3\, dx + \int \sin x\, dx$$

Use the power rule and integration of elementary functions

Using the power rule and knowledge of elementary functions, we can find the antiderivative of each term: $$F(x) = 2x^4 + C_1 - \cos x + C_2$$ Now, we combine the constants (\(C_1\) and \(C_2\)) into a single constant \(C\): $$F(x) = 2x^4 - \cos x + C$$

Use the given condition to find the constant \(C\)

The given condition states that \(F(0) = 2\), so let's plug in \(x = 0\) into our antiderivative function: $$2 = 2(0)^4 - \cos (0) + C$$ Given that \(\cos (0) = 1\), we can simplify the equation as $$2 = 1 + C$$ Now we can solve for \(C\): $$C = 1$$

Write the final antiderivative function

Inserting the value of the constant \(C\) into our antiderivative function, we get: $$F(x) = 2x^4 - \cos x + 1$$ So, the antiderivative of the given function \(f(x)\) that satisfies the given condition \(F(0) = 2\) is $$F(x) = 2x^4 - \cos x + 1$$

Key concepts

Integration

Integration is a fundamental concept in calculus, primarily used as a method for finding the total size or value, such as areas under curves, total accumulation, and more abstract quantities. For students tackling problems involving integration, it's essential to understand that integration is the inverse process of differentiation.

When you integrate a function, you're essentially looking for the antiderivative, that is, a function whose derivative gives you the initial function. In the given exercise, integration is applied to find an antiderivative for the function \(f(x) = 8x^3 + \sin x\), which represents the rate of accumulation. By integrating the function, you're finding a new function \(F(x)\) that describes the total quantity accumulated up to each point \(x\).

Power Rule

The power rule for integration is an essential tool for finding the antiderivative of functions written in the form \(x^n\), where \(n\) is any real number except -1. According to the power rule, the antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration representing any constant value that could have been differentiated away.

Applying the power rule simplifies the integration process because it provides a direct method to handle polynomial terms. For example, in our exercise, integrating \(8x^3\) using the power rule gives us \(8 \cdot \frac{x^{3+1}}{3+1}\), which simplifies to \(2x^4\).

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration, presenting them as inverse processes. It includes two parts: the first part ensures that the integral of a function over an interval can be computed using any antiderivative of the function. The second part states that the derivative of an integral of a function is equal to the original function.

In the context of the exercise, once we integrate \(f(x)\) to find \(F(x)\), we can verify that \(F(x)\) is indeed an antiderivative by differentiating it and checking if we get \(f(x)\) back. This shows the interconnectedness of differentiation and integration and solidifies the concept of antiderivatives in problem-solving.

Initial Value Problem

An initial value problem in calculus is a scenario where you are given a differential equation along with a specific value, often referred to as the initial condition, that the solution must satisfy.

Solving an initial value problem typically involves finding the general solution to a differential equation (often by integration), then using the initial condition to find the specific constant needed for a particular solution. In our exercise, after integrating to find the general form of \(F(x)\), the information that \(F(0) = 2\) serves as the initial condition. We use it to find the specific value of the integration constant \(C\), leading us to the complete and specific antiderivative of the function that satisfies the given condition.