Consider the function [tex]$f(x)=3 x^{10}+6 x^5-6 x^3-1$[/tex]. Enter an antiderivative of [tex]$f(x)$[/tex]. Do not enter +c as part of your answer.
Answer (2)
The antiderivative of the function f ( x ) = 3 x 10 + 6 x 5 − 6 x 3 − 1 can be found by integrating each term individually using the power rule. The combined antiderivative is F ( x ) = 11 3 x 11 + x 6 − 2 3 x 4 − x . The final answer is 11 3 x 11 + x 6 − 2 3 x 4 − x .
;
Find the antiderivative of each term in the function f ( x ) = 3 x 10 + 6 x 5 − 6 x 3 − 1 using the power rule for integration.
The antiderivative of 3 x 10 is 11 3 x 11 . The antiderivative of 6 x 5 is x 6 . The antiderivative of − 6 x 3 is − 2 3 x 4 . The antiderivative of − 1 is − x .
Combine the antiderivatives: F ( x ) = 11 3 x 11 + x 6 − 2 3 x 4 − x .
The antiderivative of f ( x ) is 11 3 x 11 + x 6 − 2 3 x 4 − x .
Explanation
Problem Analysis We are given the function f ( x ) = 3 x 10 + 6 x 5 − 6 x 3 − 1 and we want to find an antiderivative F ( x ) of f ( x ) . This means we need to find a function F ( x ) such that F ′ ( x ) = f ( x ) . We will use the power rule for integration, which states that ∫ x n d x = n + 1 x n + 1 + C , where n = − 1 , and C is the constant of integration.
Finding Antiderivatives of Each Term We will find the antiderivative of each term in the function separately.
The antiderivative of 3 x 10 is 3 ∫ x 10 d x = 3 ⋅ 11 x 11 = 11 3 x 11 .
The antiderivative of 6 x 5 is 6 ∫ x 5 d x = 6 ⋅ 6 x 6 = x 6 .
The antiderivative of − 6 x 3 is − 6 ∫ x 3 d x = − 6 ⋅ 4 x 4 = − 2 3 x 4 .
The antiderivative of − 1 is − ∫ 1 d x = − x .
Combining Antiderivatives Now, we combine the antiderivatives of each term to find the antiderivative of f ( x ) .
F ( x ) = 11 3 x 11 + x 6 − 2 3 x 4 − x + C
Since we are asked to not include the constant of integration, we have:
F ( x ) = 11 3 x 11 + x 6 − 2 3 x 4 − x
Final Answer Therefore, an antiderivative of f ( x ) = 3 x 10 + 6 x 5 − 6 x 3 − 1 is 11 3 x 11 + x 6 − 2 3 x 4 − x .
Examples
Understanding antiderivatives is crucial in physics, especially when dealing with motion. For example, if you know the acceleration function of a car, finding the antiderivative gives you the velocity function, and finding the antiderivative of the velocity function gives you the position function. This allows engineers to predict the car's position at any given time, which is essential for designing safe and efficient transportation systems. The same principle applies to many other areas of physics and engineering, such as calculating the flow rate of fluids or the accumulation of charge in a circuit.
