Integral Study Guide

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

((x³)/3) - x² + 3x + C

Calculate the indefinite integral: ∫ x sin x² dx

(-cos x2)/2 + C

Evaluate this indefinite integral: ∫(1+3t)t² dt

(1/3)t³ + (3/4)t⁴ +C

Evaluate this indefinite integral: ∫ √x + (1)/(2√x) dx.

(2/3)x^(3/2) + x^(1/2) + C

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

(x³)/3 + cos x + C

Evaluate the definite integral: Integration Image: https://files.catbox.moe/1m3ojd.png

(∏/2) - 1

Use substitution to evaluate the indefinite integral: ∫(xsinx²)dx

-1/2 cos x² + C

Evaluate: https://files.catbox.moe/4z4jjy.png

0

Evaluate: https://files.catbox.moe/cuesmu.png

0

Which is the trapezoidal approximation for f(x) = 1/x on [2, 5] using n=9?

0.918 General Feedback https://files.catbox.moe/f30610.PNG

Evalaute the definite integral: https://files.catbox.moe/vinw9m.PNG

1 + (√2)/2 General Feedback https://files.catbox.moe/8lmgt7.PNG

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

1.896

Use substitution to evaluate the definite integral. Integration Image: https://files.catbox.moe/6yrriv.jpg

1/2

Find the average value of f(x)=√x on the interval [1,4]

14/9

∑^100, i=1 (4+3i)

15,550

use the Fundamental Theorem of Calculus to evaluate the definite integral. Integration image: https://files.catbox.moe/0ljiy7.PNG

16

The regions A, B, and C are bounded by the graph of the function f and the x-axis. If the value of each area is 1, what is the value of Integration image: https://files.catbox.moe/yn3y55.PNG

18

Use the trapezoidal rule to approximate the area under the curve y =√x on [1, 3] using 4 subintervals.

2.793

Find F'(x) for Integration Image: https://files.catbox.moe/q27ysp.gif

2x⁻³

Use the trapezoid method to estimate the total distance traveled from t = 0 to t = 6 based on the table below. (assume v(t) > 0 from t=0 to t=6) Table https://files.catbox.moe/5hivea.PNG

35 General Feedback https://files.catbox.moe/znm0b2.PNG

Suppose evaluate Integration image: https://files.catbox.moe/e3i0f2.PNG

370

Evaluate the following sum in terms of the constant d. https://files.catbox.moe/gid9jg.PNG

71d General Feedback Where i=10, a₁₀=d In fact, for all values of i, ai=d So you have d+d+d+d+d.... from a₁₀ to a₈₀ That's 71 terms so that equals 71 d's =71d

Evaluate: Integration Image: https://files.catbox.moe/dvioo6.png

8 - 2 √2

Approximate the area under the curve f(x)=√(x+1), -1≤x≤0 with a Riemann Sum, using four sub-intervals and left endpoints.

Approximately 0.5183

Find the value of c guaranteed by the mean value theorem for integrals over [1,3] f(x)=9/x³

Approximately 1.651

Find the value of c guaranteed by the Mean Value Theorem for Integrals over [1,3] f(x)=9/(x³)

Approximately 1.651 General Feedback https://files.catbox.moe/5ged2l.PNG

Evaluate the definite integral accurate to three decimal places: Integration Image: https://files.catbox.moe/tzj8l2.gif

Approximately 57.133

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

Approximately 8.9375 General Feedback https://files.catbox.moe/4d3yqc.PNG

The rate of change of the depth of water in a water tank is given by... https://files.catbox.moe/yqemul.PNG

B General Feedback https://files.catbox.moe/8e2tdv.PNG

The rate of change of the depth of water in a water tank is given by Integration image: https://files.catbox.moe/q6evql.PNG

B (integral from 0.421 to 2.898 of r(t))

Identify: https://files.catbox.moe/bgpu94.png

Correct answer: u=x³+1, du=3x²dx

On the closed interval [1,3], which of the following could be a graph of a function with the property that... https://files.catbox.moe/dtbfbz.PNG

D General Feedback The correct answer is (D). In order for the average value of f(x) over [1,3] to equal zero, the net signed area under the curve over that interval must equal zero. Since an equal amount of area is above and below the x-axis over the interval, these areas cancel out. Therefore, the integral evaluates to zero.

Calculate the antiderivative of: cos(3x) + sin(4x) + 1

F ( x ) = [sin 3 x ]/3 − [cos4 x ]/4 + x + C

The functions: F(x)=x²/(x²+1) and G(x)=(3x²+2)/(x²+1) are antiderivatives of the same function because (choose all that apply)

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

Use substitution to evaluate the indefinite integral. Show your work and final answer for full credit. https://files.catbox.moe/m0q79n.PNG

General Feedback https://files.catbox.moe/rng0j3.PNG

Calculate the indefinite integral: ∫(1-csc t cot t) dt Show your work using the equation editor or briefly describe the steps you used to get your final answer.

General Feedback t + csc t + C

Integration Image: https://files.catbox.moe/a5chjd.gif

Integration Image: https://files.catbox.moe/qn5ebd.png

Let... Identify... Do not evaluate the integral. https://files.catbox.moe/pukwlt.PNG

Let u=2x then du=2 dx General Feedback https://files.catbox.moe/7e5yv4.PNG

Express the limit as a definite integral on the interval [a,b], where csubi, is any point in the ith subinterval. Use "Int" to stand for the integration symbol. Specify "lower limit = ___" and "upper limit = ___" Express the definite integral in the usual form in your written work to be submitted. Integration image: https://files.catbox.moe/mpysjt.PNG

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

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

None of the above General Feedback tan Θ + cos Θ + C

Calculate the indefinite integral (use a trig substitution): ∫(tan² y + 1) dy Show your work using the equation editor or briefly describe the steps you used to get your final answer.

Paraphrase and Explain: Rewrite the integral: ∫sec² y dy = tan y + C.

Integrate f(x) = x³-x on the interval [-1, 1] and explain your answer as a net area. You are required to show your work and/or provide an explanation for credit.

Paraphrase(x⁴)/4 - (x²)/2 from -1 to 1 = 0; the net area is zero because the areas are equal, one above the x-axis and one below the x-axis.

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

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

Evaluate the indefinite integral: ∫​[(x+1)√(2-x)]dx Use the html equation editor to show your work and final answer for full credit.

Paraphrase: https://files.catbox.moe/51ogv5.png

Given ∫[0,3] f(x)dx = 4 and ∫[3,6] f(x) dx = -1, find: https://files.catbox.moe/09dzc5.png

Paraphrase: https://files.catbox.moe/i961tr.png

Evaluate the definite integral: https://files.catbox.moe/z1egj5.png

Paraphrase: https://files.catbox.moe/w0gp4z.png

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).

Paraphrase: 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

The table below shows the rate in liters/min at which water leaked out of a container. Image: https://files.catbox.moe/au2iba.png Compute a right side Riemann Sum based on the table values to estimate the total amount of water that has leaked from the container.

Paraphrase: The right side sum is computed by A = (1.2)(4.3)+(1.1)(3.1)+ (1.5)(2.2) + (1.6)(1.5) = 14.27 liters

Evaluate the definite integral. Show your work and final answer for full credit. Integration Image: https://files.catbox.moe/cj6onp.png

Paraphrase: https://files.catbox.moe/0gretu.gif

Use substitution to evaluate the indefinite integral. Show your work and final answer for full credit. ∫u³√(u⁴+2)​du

Paraphrase: https://files.catbox.moe/0qzee2.gif

A speedboat travels downstream on a river. Its speed, v, in mph at certain times is given in the table above. Using a left Riemann Sum, estimate the total distance traveled by the speedboat from t = 0.5 to t = 3. Table: https://files.catbox.moe/kxy6rt.png

Paraphrase: https://files.catbox.moe/511821.bmp

Use the limit process to find the area under y = 2x on [0, 3] in the first quadrant.

Paraphrase: https://files.catbox.moe/iu4vj0.png

Integration Image: https://files.catbox.moe/a57xji.png

Paraphrase: https://files.catbox.moe/qyq15i.gif

Use the table below to estimate the total distance traveled using the Trapezoid method. (note v(t)>0 for the interval.) Table: https://files.catbox.moe/vze52x.png

Paraphrase: https://files.catbox.moe/tuolw9.png

find the upper and lower sums of y=√(1-x²) on the interval [0,1] partitioned into five subintervals of equal length Integration image: https://files.catbox.moe/04icqj.PNG

Paraphrase: https://files.catbox.moe/v6tazd.PNG

Explain if the following is always true; justify your answer: if f '(x) = g '(x) then f(x) = g(x)

Paraphrase: if f '(x) = g '(x) then f(x) = g(x) is NOT always true; f(x) and g(x) could differ by a constant.

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) = 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

Evaluate this trigonometric integral: ∫ (sin x) / (1-sin² x) dx.

sec x + C

If F(x) = ∫[0,x] sin²(2t)dt then F'(x) =

sin²(2x)

Integration Image: https://files.catbox.moe/ihle6x.jpg Let u = g(x) and du = g'(x)dx Do not evaluate the integral

u=x³+1, du=3x²dx

Evaluate this indefinite integral: ∫(x²+1) / (x²) dx.

x - x⁻¹ + C

At what values of x should the sample points be taken to get a lower sum for the integral of h on [0,6] with n=6 of the function h(x) = 3 + 2 sin x?

x=0, 1, 3, 4, 3∏/2, 5

At what values of x should the sample points be taken to get an upper sum for the integral of h on [0,6] with n=6 of the function h(x) = 3 + 2 sin x? Hint: intervals do not have to be the same width.

x=1, ∏/2, 2, 3, 4, 6

Integration image:https://files.catbox.moe/zhndjh.PNG Use substitution to evaluate the indefinite integral

∫(x²-1)³(2x)dx