Problem 63 - Calculus Early Transcendentals

Question

For the following functions \(f\), find the antiderivative \(F\) that satisfies the given condition. $$f(x)=8 x^{3}-2 x^{-2} ; F(1)=5$$

Step-by-Step Solution

Answer

Question: Find the antiderivative F(x) of the function f(x) = 8x^3 - 2x^{-2} such that F(1) = 5. Answer: The antiderivative F(x) of the given function is F(x) = 2x^4 + x^{-1} + 2.

Step 1: Find the general antiderivative G(x)

The general antiderivative \(G(x)\) is the indefinite integral of the given function \(f(x) = 8x^3 - 2x^{-2}\). To find the indefinite integral, we will use the power rule for integration, which states that: $$\int x^a dx = \frac{x^{a+1}}{a+1} + C,$$ where \(a\) is a constant and \(C\) is the constant of integration. Applying the power rule to \(f(x)\), we get: $$G(x) = \int (8x^3 - 2x^{-2}) dx = 8 \int x^3 dx - 2 \int x^{-2} dx$$ Now, we'll find the integrals of the individual terms: $$8 \int x^3 dx = 8 \left( \frac{x^{3+1}}{3+1} \right) + C_1 = 2x^4 + C_1$$ $$2 \int x^{-2} dx = -2 \left( \frac{x^{-2+1}}{-2+1} \right) + C_2 = x^{-1} + C_2$$ Therefore, the general antiderivative \(G(x)\) is given by: $$G(x) = 2x^4 + x^{-1} + C$$

Step 2: Determine the particular antiderivative F(x) using the given condition

To find the particular antiderivative \(F(x)\), we will use the given condition \(F(1) = 5\). Plug in \(x = 1\) into the expression for \(G(x)\) to solve for \(C\): $$5 = F(1) = G(1) = 2(1)^4 + (1)^{-1} + C$$ $$5 = 2 + 1 + C$$ Now, solve for \(C\): $$C = 5 - 3 = 2$$ With the value of \(C\), the particular antiderivative \(F(x)\) is given by: $$F(x) = 2x^4 + x^{-1} + 2$$

Key Concepts

Integration

Integration is a fundamental concept in calculus that allows us to find the antiderivative of a function. An antiderivative, sometimes referred to as an integral, is essentially the reverse operation of differentiation.
It helps us to determine a function whose derivative is the given function. When dealing with integration, one of the primary tasks is to compute the indefinite integral, which captures all possible antiderivatives of a function.
The indefinite integral of a function is represented with an integral sign followed by the function to be integrated and the differential, like this: \( \int f(x) \, dx \). The result of integrating a function is a new function, plus a constant \(C\), because integration is an inverse operation to differentiation. This constant \(C\), known as the constant of integration, accounts for all vertical shifts of the antiderivative graph and ensures that every possible antiderivative is considered in the general solution.

Power Rule

The power rule is a powerful technique used for both differentiation and integration. When integrating functions that are powers of \(x\), the power rule makes the process straightforward. The rule states that for any real number \(a\), except \(a = -1\), the integral of \(x^a\) with respect to \(x\) is \[ \int x^a \, dx = \frac{x^{a+1}}{a+1} + C \].
This formula comes directly from reversing the differentiation process known as the reverse power rule. When applying this rule:

• Increase the exponent by one.
• Divide by the new exponent.
• Don’t forget to add \(C\), the constant of integration.

In the given exercise with \(f(x) = 8x^3 - 2x^{-2}\), we used the power rule separately for both terms:

• For \(8x^3\), the integral becomes \(2x^4 + C_1\).
• For \(-2x^{-2}\), the integral is \(x^{-1} + C_2\).

Thus, the general antiderivative includes these terms adjusted by the power rule with a single consolidation constant \(C\).

Initial Conditions

After finding the general antiderivative, it's often necessary to determine a specific solution called a particular antiderivative. This is where initial conditions come in handy.
Initial conditions are particular values that the antiderivative must satisfy. They allow us to solve for the constant of integration \(C\) that appears in the indefinite integral.
In this context, you apply the known condition(s) to the general antiderivative to find the exact value of \(C\). Using our example, the initial condition provided was \(F(1) = 5\).
To find the particular antiderivative, we substituted \(x = 1\) into the general antiderivative formula \(G(x) = 2x^4 + x^{-1} + C\) and set it equal to 5:

• Substituting gives \(5 = 2(1)^4 + (1)^{-1} + C\).
• This simplifies to \(5 = 3 + C\).
• Solving for \(C\) yields \(C = 2\).

Thus, the specific antiderivative satisfying the initial conditions is \(F(x) = 2x^4 + x^{-1} + 2\), which uniquely meets the initial condition \(F(1) = 5\).