Q: Which of the following is the correct power rule for integration?

$$\int x^n\,dx = \frac{x$$^{n+1}$$}{n+1} + C$$, for $$n \neq -1$$

Question 1

Which of the following best describes the relationship between the functions $$f(x)$$, $$g(x)$$, and $$h(x) as $$shown in the image, given that their tangent lines at $$x = a$$ all have the same slope?

Accepted Answer

They are all antiderivatives of the same function, differing only by a constant.

Question 2

Given that $$f(x)$$, $$g(x)$$, and $$h(x)$$ have the same derivative at every point, what can be concluded about their general form?

Accepted Answer

They must all differ by a constant, i.e., $$f(x) = g(x) + C_1$ and h(x) = g(x$$) + $$C_2.$$

Question 3

If $$F(x) is an $$antiderivative of $$f(x)$$, which of the following is also an antiderivative of $$f(x)$$?

Accepted Answer

$$F(x) + C$$, where $$C is $$any constant

Question 4

Suppose $$f(x) = 2x. $$Which of the following is a general antiderivative of $$f(x)$$?

Accepted Answer

$$x^2 + C$

Question 5

A car's velocity is given by $$v(t) = 3t^2$. If the car starts from rest at t = 0$, what is the position function s(t$$)$$?

Accepted Answer

$$s(t) = t^3$

Question 6

Which of the following statements about antiderivatives is correct?

Accepted Answer

All antiderivatives of a function differ by a constant.

Question 7

If $$\frac{d}{dx}F(x) = f(x)$$, what is the indefinite integral of $$f(x)$$?

Accepted Answer

$$\int f(x)\,dx = F(x) + C$$

Question 8

Which rule allows you to integrate $$af(x) + bg(x)$$, where $$a$$ and $$b$$ are constants?

Accepted Answer

Constant Multiple Rule and Sum or Difference Rule

Question 9

What is the antiderivative of $$f(x) = \frac{1}{x}$$ for $$x \u003e 0$$?

Accepted Answer

$$\ln|x| + C$$

Question 10

A medication is administered at a rate of $$r(t) = 5e$$^{0.2t}$$ mg/hour. What is the total amount of medication administered from t = 0$ to t = 5$ hours?

Accepted Answer

$$\$$int_0$$^5 5e$$^{0.2t}$$\,dt = 25(e$$^{1}$$ - 1)$$

Question 11

If $$\int f(x)\,dx = x^3$ + 2x + C$$ and $$f(1) = 5$$, what is the value of $$C$$?

Accepted Answer

$$C = 2$$

Question 12

Which of the following is the correct power rule for integration?

Accepted Answer

$$\int x^n\,dx = \frac{x$$^{n+1}$$}{n+1} + C$$, for $$n 
eq -1$$

Question 13

A population of bacteria grows at a rate $$P'(t) = 4t + 2. If $$the initial population at $$t = 0 is 100$$, what is the population at time $$t$$?

Accepted Answer

$$P(t) = 2t^2$ + 2t + 100$$

Question 14

Which of the following is the general antiderivative of $$f(x) = \cos(x)$$?

Accepted Answer

$$\sin(x) + C$$

Question 15

If $$\int f(x)\,dx = F(x) + C$$ and $$F(2) = 7$$, $$F(0) = 3$$, what is $$\$$int_0$$^2 f(x)\,dx$$?

Accepted Answer

$$4$$

Antiderivatives and Indefinite Integration

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