Problem 12 - Calculus Early Transcendentals

Question

Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. $$g(x)=11 x^{10}$$

Step-by-Step Solution

Answer

Question: Find the antiderivative of the function g(x) = 11x^10, and check whether your answer is correct by differentiation. Answer: The antiderivative of the given function g(x) = 11x^10 is F(x) = x^11 + C, where C is the constant of integration. After differentiating F(x), we get back the original function g(x) = 11x^10, which confirms that our antiderivative is correct.

Step 1: Identify the power rule for integration

The power rule for integration states that the integral of a function f(x) = x^n, where n is a constant, is given by: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ Here, C is the constant of integration.

Step 2: Apply the power rule to the given function

Given function is: $$g(x)=11 x^{10}$$ Using the power rule for integration, we get the antiderivative F(x) as: $$F(x) = \int 11 x^{10} dx = 11\int x^{10} dx = 11\cdot\frac{x^{10+1}}{10+1} + C = \frac{11x^{11}}{11} + C$$ where C is the constant of integration.

Step 3: Simplify the antiderivative

Simplifying the antiderivative F(x), we get: $$F(x) = x^{11} + C$$

Step 4: Check the answer by differentiation

To check if our antiderivative is correct, we differentiate F(x) with respect to x: $$\frac{d}{dx}(x^{11}+C)=11x^{10}$$ The result matches the given function g(x), which means: 1. our antiderivative is correct 2. The original function was indeed the derivative of the antiderivate function we found.

Key Concepts

Power Rule for Integration

Understanding the power rule for integration is essential for students tackling calculus problems. It is remarkably straightforward: when you integrate a monomial—a single term in the form of \( x^n \), where \( n \) is a real number, you can apply this rule. Here's the recipe: increase the exponent by one, then divide the result by the new exponent, and don't forget to add the constant of integration, \( C \), at the end.

For example, taking the integral of \( g(x) = 11x^{10} \) would require these steps: first, increase the exponent 10 to 11 to obtain \( x^{11} \); then, divide by the new exponent 11, making the coefficient \( \frac{11}{11} \) (which simplifies to 1); finally, tack on \( C \). Thus, the antiderivative of \( g(x) \) is \( F(x) = x^{11} + C \).

Exercise Improvement Tip:

It is beneficial to visualize this process with a function where \( n \) is not an integer to see how the rule applies universally. For instance, \( \( \'int x^{3/2} dx = \frac{2}{5}x^{5/2} + C \) \) illustrates that the power rule seamlessly extends to fractional exponents.

Constant of Integration

Every calculus student soon learns the mantra: 'Don't forget to add \( C \)!' But why do we add this constant of integration when finding antiderivatives? The reason lies in the fact that differentiation wipes away constants—since the derivative of a constant is zero. As a result, when we reverse the process (integrate), we need to acknowledge that there could have been any constant value there initially.

Technically, for any continuous function like our \( g(x) = 11x^{10} \) example, there are infinite antiderivatives, each differing by a constant. When we state \( F(x) = x^{11} + C \), we cover all scenarios, encompassing every possible constant. Add any real number for \( C \)—be it 47, -3, or \( \( pi \) \)—you've nabbed a valid antiderivative. This also highlights the importance of initial conditions or boundaries when solving specific problems to pinpoint the exact value of \( C \).

Checking Antiderivatives by Differentiation

After finding an antiderivative, it is prudent to verify correctness. Fortunately, we have a built-in 'quality check' method—differentiation. Since antiderivatives are, by definition, the reverse of derivatives, the derivative of an antiderivative should yield the original function.

Following our steps for the function \( g(x) = 11x^{10} \), we proposed \( F(x) = x^{11} + C \) as the antiderivative. To check, we differentiate \( F(x) \) and, as expected, we obtain: \( \frac{d}{dx}(x^{11}+C)=11x^{10} \). This derivative matches our original function \( g(x) \), thus confirming the accuracy of our antiderivative.

Exercise Improvement Tip:

It is useful for students to tackle functions that result in less obvious derivatives for practice, such as \( e^{x} \) or \( sin(x) \). The act of verifying these kinds of antiderivatives reinforces understanding of both differentiation and how it relates to integration.