Question 1

Which of the following is the correct antiderivative of $$\int \frac{1}{x} \, dx$$?

Accepted Answer

$$\ln|x| + C$$

Question 2

Given $$\int e$$^{2x}$$ \, dx$$, which substitution leads to the correct antiderivative?

Accepted Answer

Let $$u = 2x$$, then $$du = 2dx$$

Question 3

Which of the following is the general solution to the differential equation $$\frac{dy}{dx} = 3x^2$?

Accepted Answer

$$y = x^3$ + C$$

Question 4

If $$f'(x) = \cos x$$, what is $$f(x)$$?

Accepted Answer

$$\sin x + C$$

Question 5

Evaluate $$\int (2x^3$ - 5x^2$ + 4) \, dx.$$

Accepted Answer

$$\frac{1}{2}x^4$ - \frac{5}{3}x^3$ + 4x + C$$

Question 6

A ball is thrown upward with velocity $$v(t) = 32 - 32t. $$What is the position function $$s(t) if$$ $$s(0) = 0$$?

Accepted Answer

$$s(t) = 32t - 16t^2$

Question 7

Which property of indefinite integrals is illustrated by $$\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx$$?

Accepted Answer

Linearity

Question 8

Solve the initial value problem: $$\frac{dy}{dx} = 2x$$, $$y(0) = 5.$$

Accepted Answer

$$y = x^2$ + 5$$

Question 9

Which of the following is the antiderivative of $$\sec^2$ x$$?

Accepted Answer

$$	an x + C$$

Question 10

A medication is administered such that its rate of absorption is $$A'(t) = 5e$$^{-0.2t}$$. If A(0) = 0$, what is the total amount absorbed after t$ hours?

Accepted Answer

$$A(t) = -25e$$^{-0.2t}$$ + 25$$

Question 11

Which of the following is the correct antiderivative for $$\int \sin x \, dx$$?

Accepted Answer

$$-\cos x + C$$

Question 12

If $$\int f(x) \, dx = F(x) + C$$, which of the following statements is true?

Accepted Answer

$$F'(x) = f(x)$$

Question 13

Evaluate $$\int (3e^x + 2\cos x) \, dx.$$

Accepted Answer

$$3e^x + 2\sin x + C$$

Question 14

A car's acceleration is $$a(t) = 6t. If $$the initial velocity is $$v(0) = 2$$, what is the velocity function $$v(t)$$?

Accepted Answer

$$v(t) = 3t^2$ + 2$$

Question 15

Which of the following is the antiderivative of $$\frac{1}{1 + x^2$}$$?

Accepted Answer

$$\arctan x + C$$

Question 16

If $$\int f(x) \, dx = F(x) + C$$, what is $$\int 5f(x) \, dx$$?

Accepted Answer

$$5F(x) + C$$

Question 17

A population grows at a rate $$P'(t) = 0.1P(t). If$$ $$P(0) = 100$$, what is the general solution for $$P(t)$$?

Accepted Answer

$$P(t) = 100e$$^{0.1t}$$

Antiderivatives and Initial Value Problems

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