Please read the Fundamental Theorem of Calculus Part 2 (Theorem 1.5).

In this problem we work through the steps of evaluating the definite integral ∫
9
1
5

4
√
x3
dx.

a) Find an antiderivative of f(x)=
5

4
√
x3
. (Re-read section 4.10 from MAT135
Links to an external site.
if you are unsure.)

An antiderivative of f is: F(x)=
[ Select ]
-(5/2) x^(-1/2)
-(5/4) x^(-1/2)
-(1/2) x^(-5/2)
-(5/2) x^(-5/2)

b) What is the final answer? The definite integral ∫
9
1
5

4
√
x3
dx =
[ Select ]
3/5
5/4
(5/4)^3 - 1
5/3
1 - (5/4)^3
-5/4
-5/3

(Use a simple calculator for (b) if you want.)

Question

Please read the Fundamental Theorem of Calculus Part 2 (Theorem 1.5).

In this problem we work through the steps of evaluating the definite integral ∫
9
1
5

4
√
x3
dx.

a) Find an antiderivative of  f(x)=
5

4
√
x3
. (Re-read section 4.10 from MAT135
Links to an external site.
 if you are unsure.)

An antiderivative of f is: F(x)= 
[ Select ]
-(5/2) x^(-1/2)
-(5/4) x^(-1/2)
-(1/2) x^(-5/2)
-(5/2) x^(-5/2)

b) What is the final answer? The definite integral ∫
9
1
5

4
√
x3
dx = 
[ Select ]
3/5
5/4
(5/4)^3 - 1
5/3
1 - (5/4)^3
-5/4
-5/3

(Use a simple calculator for (b) if you want.)

BlackTom AI · Accepted Answer

Step into the problem by parsing the integrand. The expression suggests f(x) = (5/4) x^(-3/2) because the antiderivative options involve x^(-1/2) terms upon integrating x^(-3/2). Now analyze each option for part (a):
- Option 1: -(5/2) x^(-1/2). If you differentiate this, you get -(5/2) * (-1/2) x^(-3/2) = (5/4) x^(-3/2), which matches f(x). This makes it a correct antiderivative.
- Option 2: -(5/4) x^(-1/2). Differentiating yields -(5/4) * (-1/2) x^(-3/2) = (5/8) x^(-3/2), which does not equal f(x). This one is incorrect.
- Option 3: -(1/2) x^(-5/2). Differentiating gives -(1/2) * (-5/2) x^(-7/2) = (5/4) x^(-7/2), which is not our target form. This is incorrect.
- Option 4: -(5/2) x^(-5/2). Differentiating yields -(5/2) * (-5/2) x^(-7/2) = (25/4) x^(-7/2), which again does not match f(x). This is incorrect.
So, for (a), only the first option behaves as a valid antiderivative of f(x) = (5/4) x^(-3/2).

For part (b), use the Fundamental Theorem of Calculus with F(x) being an antiderivative chosen consistently with part (a): F(x) = -(5/2) x^(-1/2). Evaluate a definite integral from x = 1 to x = 9 as F(9) - F(1):
- F(9) = -(5/2) * 9^(-1/2) = -(5/2) * (1/3) = -5/6.
- F(1) = -(5/2) * 1^(-1/2) = -(5/2) * 1 = -5/2.
Compute the difference: F(9) - F(1) = (-5/6) - (-5/2) = (-5/6) + (5/2) = (-5/6) + (15/6) = 10/6 = 5/3.
Now compare against the provided choices for (b):
- 3/5: This does not match the computed 5/3; it would correspond to an incorrect sign or exponent interpretation. This is incorrect.
- 5/4: This value does not equal the definite integral result and would imply a different antiderivative or evaluation. This is incorrect.
- (5/4)^3 - 1: This is a algebraic expression unrelated to the evaluated definite integral from 1 to 9; it’s incorrect.
- 5/3: This matches the computed value from the FTC calculation, so this option is correct.
- 1 - (5/4)^3: This again is not aligned with the evaluation and is incorrect.
- -5/4: This is a simple constant not matching the integral result; incorrect.
- -5/3: The negative of the correct value; incorrect.
In summary, the first part’s correct antiderivative corresponds to the first option, and the definite integral result matches the 5/3 option in part (b).

类似问题

Find the value of \( \int_{0}^{4}({-x+3\sqrt{x}}){dx}\)

Which of the following is equal to ∫ − 𝜋 𝜋 𝑓 ( 𝑥 ) 𝑑 𝑥 correct to 3 decimal places, where 𝑓 ( 𝑥 ) = { − 4 𝑥 if − 𝜋 ≤ 𝑥 ≤ 0 3 − 3 cos ⁡ 𝑥 if 0 < 𝑥 ≤ 𝜋 .

Which of the following is equal to ∫ − 𝜋 𝜋 𝑓 ( 𝑥 ) 𝑑 𝑥 correct to 3 decimal places, where 𝑓 ( 𝑥 ) = { − 7 𝑥 if − 𝜋 ≤ 𝑥 ≤ 0 sin ⁡ 𝑥 if 0 < 𝑥 ≤ 𝜋 .

You are given that \[ \int _{-6}^{-5} f(x) \, dx = -5, \quad \int _{-6}^{0} f(x) \, dx = 7, \quad \text {and} \quad \int _{0}^{-5} g(x) \, dx = -1. \] What is \[ \int _{-5}^{0} \left [ f(x) + 5 g(x) \right ] \, dx \, ? \]

你被赋予 \[ \int _{-2}^{-1} f（x） \， dx = -3， \quad \int _{-2}^{6} f（x） \， dx = -4， \quad \text {and} \quad \int _{6}^{-1} g（x） \， dx = 1. \] 什么是 \[ \int _{-1}^{6} \left [ f（x） + 4 g（x） \right ] \， dx \，？ \]

You are given that ∫ − 4 − 3 𝑓 ( 𝑥 ) 𝑑 𝑥 = 2 , ∫ − 4 6 𝑓 ( 𝑥 ) 𝑑 𝑥 = 4 , and ∫ 6 − 3 𝑔 ( 𝑥 ) 𝑑 𝑥 = 1. What is ∫ − 3 6 [ 𝑓 ( 𝑥 ) + 3 𝑔 ( 𝑥 ) ] 𝑑 𝑥 ?

You are given that ∫ 1 2 𝑓 ( 𝑥 ) 𝑑 𝑥 = 5 , ∫ 1 7 𝑓 ( 𝑥 ) 𝑑 𝑥 = 1 , and ∫ 7 2 𝑔 ( 𝑥 ) 𝑑 𝑥 = 3. What is ∫ 2 7 [ 𝑓 ( 𝑥 ) + 3 𝑔 ( 𝑥 ) ] 𝑑 𝑥 ?

你被赋予 ∫ −1 −2 f（x）\，dx=−3，∫ 6 −2 f（x）\，dx=−4，且∫ −1 6 g（x）\，dx=1。什么是 ∫ 6 −1 [f（x）+4g（x）]\，dx\，？

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