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Chapter 4: Problem 12

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

Short Answer

Expert verified
Answer: G(x) = x^11 + C

Step by step solution

01

Identify the function to integrate

We are given the function $$g(x) = 11x^{10}$$, and we need to find its antiderivative.
02

Apply the power rule for integration

To integrate the function $$11x^{10}$$, we will use the power rule for integration, which states that: $$\int x^n dx = \frac{1}{n+1}x^{n+1} + C$$ where n is a constant, and C is the constant of integration. In our case, n = 10.
03

Integrate the function

Now, let's apply the power rule to our given function $$g(x) = 11x^{10}$$: $$\int 11x^{10} dx = 11\int x^{10} dx = 11\frac{1}{10+1}x^{10+1} + C = \frac{11}{11}x^{11} + C = x^{11} + C$$ So, the antiderivative of $$g(x) = 11x^{10}$$ is: $$G(x) = x^{11} + C$$
04

Check the result by taking the derivative

Now, let's take the derivative of our antiderivative and verify that it matches the original function: $$G'(x) = \frac{d}{dx}(x^{11} + C) = 11x^{10}$$ Hence, the derivative of our antiderivative $$G(x)$$ is equal to the original function $$g(x)$$: $$G'(x) = g(x) = 11x^{10}$$ Our antiderivative is correct.
05

Final Answer

The antiderivative of the given function $$g(x) = 11x^{10}$$ is: $$G(x) = x^{11} + C$$

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