AP CALC

Evaluate this indefinite integral: ∫x2-2x+3 dx.

((x3)/3) - x2 + 3x +C

Calculate the indefinite integral: ∫ x sin x 2 dx

(-cos x2)/2 + C

Find the indefinite integral: ∫(√y)y2 dy

(2/7) y7/2 + C

Evaluate this trigonometric integral: ∫ x2 - sin x dx.

(x^3)/3 + cosx + C

Determine the area of the indicated region accurate to the three decimal places: y= x+ sin x

(∏2/2)+2

Which is the correct derivative of y=\ln(\frac{e^x}{e^x-1})y=ln(ex−1ex​)?

-1/ (e^x -1)

Evaluate the definite integral: f 1 (4t^3 -2t) dt -1

0

Using a left Riemann Sum, what is the approximation for the area under y=2x-x² from x=1 to x=2 using four equal subintervals?

0.781

Using the trapezoidal rule, estimate the area under the curve y = sinx [0, π] using 4 subintervals.

1.896

Integrate: \int_{ }^{ }xe^{x^2}dx∫​xex^2dx

1/2ex^2+C

Use substitution to evaluate the definite integral. \int_0^1\left(x\sqrt{1-x^2}\right)dx∫01​(x1−x2​)dx

1/3

Evaluate \int\frac{x^2}{\sqrt{1-x^6}}dx∫1−x^6​x2​dx

1/3 arc sin (x^3)+C

If f(x)=x^3-3x^2+8x+5f(x)=x3−3x2+8x+5 and g(x)=f^{-1}(x)g(x)=f−1(x), then g'(5) is

1/8

A table of values for a continuous function f is shown below. If four equal subintervals of [0, 2] are used, which of the following is the trapezoidal approximation of \int_0^2f(x)\operatorname dx∫02​f(x)dx?

12

If you were to use 3 midpoint rectangles of equal length to approximate the area under the curve of f(x)=x2+2 from x = 0 to x = 3, how close would the approximation be to the exact area under the curve?

14

Find F(x)=\int_\frac{\mathrm\pi}2^{x^2}\sqrt{\sin t}\operatorname dtF(x)=∫2π​x2​sint​dt

2x sqrt(sin x^2)

Find F'(x) for F(x) = f x^2 (1/t^2)dt 2

2x^-3

Find the value of x at which the function y=x² reaches its average value on the interval [0,10].

5.774

Find the limit of s(n) as n-> infinity s(n)= 64/n^3

64/3

Find the volume of the solid obtained by rotating the region bounded by y=(1/x), x=(1/2), x=4 and the x axis about the y axis.

7π

Evaluate: f 8 sqrt(2/x) dx 1

8 - 2 sqrt(2)

A stone is thrown straight up from the top of a building with initial velocity 40 ft/sec and hits the ground 4 seconds later. What is the height of the building, in feet?

96

Approximate the area under the curve f(x)=x2+2, -2≤x≤1 with a Riemann Sum, using six sub-intervals and right endpoints.

Approximately 8.375

Which of the following statements is/are true about integrals and integration:

Both A and B.

The functions F(x)=\frac{x^2}{x^2+1}F(x)=x2+1x2​ and G(x)=\frac{3x^2+2}{x^2+1}G(x)=x2+13x2+2​ are antiderivatives of the same function because (choose all that apply)

F '(x) = G '(x) G(x)dx - F(x)dx = 2

Which of the following are true? I. If f(x) is increasing on [a, b], then the left Riemann sum for \int_a^bf(x)\operatorname dx∫ab​f(x)dx is an underestimate. II. if f(x) is increasing on [a,b] and g(x) is decreasing on [a,b] then the left Riemann sum \int_a^bf(x)\operatorname dx>\int_a^bg(x)\operatorname dx∫ab​f(x)dx>∫ab​g(x)dx III. If f(x) is concave down on [a,b] then the midpoint Riemann sum for \int_a^bf(x)\operatorname dx∫ab​f(x)dx is an overestimate.

I and III

Express the limit as a definite integral on the interval [a, b] , where ci is any point on the ith subinterval.

Lower limit 0, Upper limit 4, ∫6x(4-x2)dx

Find the indefinite integral: ∫(sec2 Θ - sin Θ) dΘ

None of the above

Use substitution to evaluate the indefinite integral: f(x^2-1)^3 (2x)dx

[(x^2-1)^4]/4 + C

For which of the following constants a and b is the function F(x) = axsinx - (x² - b)cosx an antiderivative of f(x)= x²sinx?

a = 2, b = 2

Solve the differential equation: f''(x)=x^2;\;f'(0)=6;\;f(0)=3f′′(x)=x2;f′(0)=6;f(0)=3

f'(x)=(1/3)x^3+c, 6 = c, f'(x)=(1/3)x^3+6, f(x)=(1/12)x^4+6x+d, d = 3, so f(x) = (1/12)x^4 + 6x + 3.

find f(x) where f(2) = 3 and f '(x) = 4x + 5

f(x) = 2x² + 5x + C f(2) = 3, thus 3 = 2(2)² + 5(2) + C 3 = 18 + C f(x) = 2x² + 5x - 15

Use the limit process to find the area of the region between the graph y = 3x-4 and the x-axis on the interval [2,5].

limn→∞​∑i=1n​[3(2+3i/n​)−4](3/n​)

A particle moves along a scale with velocity v = 3t + 7. If the particle is at 4 on the scale at time t = 1, find the position function s(t).

s(t) is an antiderivative of v(t) = 3t + 7 s(t) = 3/2 t² + 7t + C 4 = 3/2(1)² + 7(1) + C C = -9/2 s(t) = 3/2t² + 7t - 9/2

Solutions of the differential equation y dy = x dx are in which form?

x^2 -y^2 = C

Solve the differential equation: \frac{dy}{dx}=\frac{y}{2\sqrt{x}}dxdy​=2x​y​ for the point (4, 1)

y = e^sqrtx - 2

The slope field shown below matches which differential equation? #10

y' = 2x-4

The region bounded by y = 0 and y=3x-x2 is rotated about the x-axis. Which of the following will give the correct volume of the solid that is generated?

π∫03​(9x^2−6x^3+x^4)dx

The area bounded by the parabola y=2x² and the line y=2x+4 is given by which of the following?

∫−1 2​(4+2x−2x^2)dx