Integrals Retake

 $\int_{ }^{ }\left(x^3-3x\right)dx$  =

•  $3x^2-3+C$  

•  $4x^4-6x^2+C$  

•  $\frac{x^4}{3}-3x^2+C$  

•  $\frac{x^4}{4}-3x^{ }+C$  

•  $\frac{x^4}{4}-\frac{3x^2}{2}+C$  

The graph of a piecewise-linear function, f, for -1 < x < 4, is shown above. What is the value of $\int_{-1}^4\ f\left(x\right)\ dx$  ?

• 1

• 2.5

• 4

• 5.5

• 8

The values of a continuous function f for selected values of x are given in the table. What is the value of the left Riemann sum approximation to $\int_0^{50}f\left(x\right)\ dx$  using the subintervals [0, 25], [25, 30], and [30, 50]?

• 290

• 360

• 380

• 390

• 430

A particle moves along the x-axis. The velocity of the particle at time t is $6t-t^2$ .  What is the total distance traveled by the particle from time t = 0 to t = 3? 

• 3

• 6

• 9

• 18

• 27

 $\int_{ }^{ }\ \frac{x\ dx}{\sqrt{3x^2+5}}$  

•  $\frac{1}{9}\left(3x^2+5\right)^{\frac{3}{2}}+C$  

•  $\frac{1}{4}\left(3x^2+5\right)^{\frac{3}{2}}+C$  

•  $\frac{1}{12}\left(3x^2+5\right)^{\frac{1}{2}}+C$  

•  $\frac{1}{3}\left(3x^2+5\right)^{\frac{1}{2}}+C$  

•  $\frac{3}{2}\left(3x^2+5\right)^{\frac{1}{2}}+C$  

If $\frac{dy}{dx}=\ \tan\ x$ , then  $y\ =\$  

•  $\frac{1}{2}\tan^2x\ +\ C$  

•  $\sec^2x\ +\ C$  

•  $\ln\left|\sec\ x\right|\ +\ C$  

•  $\ln\left|\cos\ x\right|\ +\ C$  

•  $\sec\ x\ \tan\ x\ +\ C$  

 $\int_1^4\ \left|x-3\right|dx$  

•  $-\frac{3}{2}$  

•  $\frac{3}{2}$  

•  $\frac{5}{2}$  

•  $\frac{9}{2}$  

•  $5$  

The average value of $\sqrt{x}$ over the interval 0 < x < 2 is 

•  $\frac{1}{3}\sqrt{2}$  

•  $\frac{1}{2}\sqrt{2}$  

•  $\frac{2}{3}\sqrt{2}$  

• 1

•  $\frac{4}{3}\sqrt{2}$  

 $\int_{ }^{ }\sin\left(2x+3\right)dx$  

•  $\frac{1}{2}\cos\left(2x+3\right)+C$  

•  $\cos\left(2x+3\right)+C$  

•  $-\cos\left(2x+3\right)+C$  

•  $-\frac{1}{2}\cos\left(2x+3\right)+C$  

•  $-\frac{1}{5}\cos\left(2x+3\right)+C$  

 $\int_0^1\left(3x-2\right)^2dx$  

•  $-\frac{7}{3}$  

•  $-\frac{7}{9}$  

•  $\frac{1}{9}$  

•  $1$  

•  $3$