Find the general antiderivative of [tex]f(x)=6 x^2+\frac{2}{x}+\frac{3}{x^6}-\sqrt{x}[/tex]
Answer (2)
To find the general antiderivative of the function f ( x ) = 6 x 2 + x 2 + x 6 3 − x , we rewrite it in terms of powers of x and integrate each term using the power rule. The result is F ( x ) = 2 x 3 + 2 ln ∣ x ∣ − 5 x 5 3 − 3 2 x 3/2 + C .
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Rewrite the function as f ( x ) = 6 x 2 + 2 x − 1 + 3 x − 6 − x 1/2 .
Integrate each term using the power rule: ∫ x n d x = n + 1 x n + 1 + C and ∫ x 1 d x = ln ∣ x ∣ + C .
Obtain the antiderivatives: ∫ 6 x 2 d x = 2 x 3 , ∫ 2 x − 1 d x = 2 ln ∣ x ∣ , ∫ 3 x − 6 d x = − 5 x 5 3 , and ∫ − x 1/2 d x = − 3 2 x 3/2 .
Combine the results with the constant of integration: 2 x 3 + 2 ln ∣ x ∣ − 5 x 5 3 − 3 2 x 3/2 + C .
Explanation
Problem Analysis We are asked to find the general antiderivative of the function f ( x ) = 6 x 2 + f r a c 2 x + f r a c 3 x 6 − s q r t 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 in t x n d x = f r a c x n + 1 n + 1 + C , where n n e q − 1 , and the fact that in t f r a c 1 x d x = l n ∣ x ∣ + C .
Rewriting the Function First, rewrite the function in terms of powers of x : f ( x ) = 6 x 2 + 2 x − 1 + 3 x − 6 − x 1/2 . Now, we integrate each term separately.
Integrating the first term Integrating 6 x 2 : in t 6 x 2 d x = 6 in t x 2 d x = 6 c d o t f r a c x 2 + 1 2 + 1 = 6 c d o t f r a c x 3 3 = 2 x 3
Integrating the second term Integrating 2 x − 1 : in t 2 x − 1 d x = 2 in t f r a c 1 x d x = 2 l n ∣ x ∣
Integrating the third term Integrating 3 x − 6 : in t 3 x − 6 d x = 3 in t x − 6 d x = 3 c d o t f r a c x − 6 + 1 − 6 + 1 = 3 c d o t f r a c x − 5 − 5 = − f r a c 3 5 x − 5 = − f r a c 3 5 x 5
Integrating the fourth term Integrating − x 1/2 : in t − x 1/2 d x = − in t x 1/2 d x = − f r a c x 1/2 + 1 1/2 + 1 = − f r a c x 3/2 3/2 = − f r a c 2 3 x 3/2
Combining the results Combining the results and adding the constant of integration C , we get: F ( x ) = 2 x 3 + 2 l n ∣ x ∣ − f r a c 3 5 x 5 − f r a c 2 3 x 3/2 + C
Final Answer Therefore, the general antiderivative of f ( x ) is 2 x 3 + 2 l n ∣ x ∣ − f r a c 3 5 x 5 − f r a c 2 3 x 3/2 + C .
Examples
Understanding antiderivatives is crucial in physics, especially when dealing with motion. For example, if you know the acceleration of an object as a function of time, finding the antiderivative gives you the velocity function. Then, taking the antiderivative of the velocity function gives you the position function. This allows engineers to predict the trajectory of projectiles or design smoother rides by controlling acceleration.
