Calculus 2 - Integrals

Evaluate the integral ∫ (2+2y)² dy on [2,5]

252

Evaluate the integral by making the given substitution. ∫ x³(7 + x⁴)⁶ dx, u = 7 + x⁴

(1/28)(7 + x⁴)⁷ + C

Evaluate the indefinite integral ∫ [(ln x)⁵⁰]/x dx

(1/51)(ln x)⁵¹ + C

Evaluate the indefinite integral ∫ e^(x)√(43 + e^(x))

(2/3)(43 + e^(x))^(3/2) + C

Evaluate the integral by making the given substitution. ∫ x²√(x³ + 37) dx, u = x³ + 37

(2/9)(x³ + 37)^(3/2) + C

Evaluate the integral ∫ e^(2θ) sin 3θ dθ

(4/26)e^(2θ) sin(3θ) - (3/4)e^(2θ) cos(3θ) + C

Evaluate the integral ∫ 5p²ln(p) dp

(5/3)p³ln(p) - (5/9)p³ + C

Evaluate the integral ∫ e^(8θ) sin(9θ) dθ

(8/145)e^(8θ)sin(9θ) - (9/145)e^(8θ)cos(9θ) + C

Evaluate the integral using integration by parts with the indicated choices of u and dv: ∫ 8x² ln x dx; u = ln x, dv = 8x² dx

(8/3)x³lnx - (8/9)x³ + C

Evaluate the integral ∫ 9x cos 8x dx

(9/8)xsin8x + (9/64)cos8x + C

Evaluate the integral by making the given substitution. ∫ cos¹⁷θsinθ dθ, u = cos θ

-(1/18)cos¹⁸x + C

Evaluate the integral ∫ t sin 8t dt

-(1/8)t cos(8t) + (1/64)sin(8t) + C

The velocity function (in meters per second) is given for a particle moving along a line. v(t) = t² - 2t - 8, 1 ≤ t ≤ 5

-(44/3)

Evaluate the integral ∫ 9xe^(-7x) dx

-(9/7)xe^(-7x) - (9/49)e^(-7x) + C

Evaluate the integral ∫ 9x² sin πx dx

-(9/π)x²cos(πx) + (18/π²)xsin(πx) + (18/π³)cos(πx) + C

First make a substitution and then use integration by parts to evaluate the integral. ∫ 9θ³ cos(θ²) dθ on [√(π/2) , √π]

-(9π/4) - (9/2)

Evaluate the integral ∫ (y -3)(2y + 1) dy on [0, 3]

-13.5

Evaluate the integral ∫ (9x² + 7)e^(-x) dx on [0, 1]

-16e⁻¹

Evaluate ∫ (t+2)(3t-4) dt on [0, 2]

-4

First make a substitution and then use integration by parts to evaluate the integral. ∫ t³e^(-t²) dt

-½t²e^(-t²) - ½e^(-t²) + C

Evaluate the integral ∫ (x - 7) / √x dx on [1, 16]

0

Evaluate the integral ∫ lny/√y dy on [25, 36]

12ln36 - 10ln25 - 4

Evaluate the integral by interpreting it in terms of areas. ∫ (4 - 2x) dx, on [-3, 5]

16

If ƒ is integrable on [a, b] the following equation is correct. ∫ (x² + 4x - 7) dx on [1, 7]

168

Evaluate the integral ∫ 16 ln³√x dx

16xln³√x - (16/3)x + C

If ƒ is integrable on [a, b] the following equation is correct. ∫ (1 + 2x) dx on [-1, 5]

30

Evaluate ∫ 3e^x - 5cos x dx on [0, π/2]

3e∧(π/2) - 8

Evaluate the integral ∫ 7re^(r/6) dr

42re^(r/6) - 252e^(r/6)

Find the most general antiderivative of the function. f(x) = 9√x + 2cosx

F(x) = (18/3)x^(3/2) + 2sinx + C

Find the most general antiderivative of the function. f(x) = 9x^(1/4) - 3x^(3/4)

F(x) = (36/5)x^(5/4) - (12/7)x^(7/4) + C

Find the most general antiderivative of the function. f(x) = 8/x⁷

F(x) = -(4/3)x⁻⁶ + C

Find the most general antiderivative of the function. f(x) = 5e^x + 4sec²x

F(x) = 5e^x + 4tanx + C

Find the most general antiderivative of the function. f(x) = ¾ + (2/3)x² - (5/6)x³

F(x)= (3/4)x + (2/9)x³ - (5/24)x⁴ + C

Find the Riemann sum for f(x) = 2sinx, 0 ≤ x ≤ 3π/2, with six terms, taking the sample points to be right endpoints. Round your answer to six decimal places.

R₆ = 1.110721

The velocity function v(t) (in meters per second) is given for a particle moving along a line. v(t) = 3t - 7, 0 ≤ t ≤ 4 a) find the displacement d₁ traveled by the particle during the time interval given above. b) find the total distance d₂ traveled by the particle during the time interval given above.

a) -4 m b) (37/3) m

Evaluate the integral ∫ e^(u + 1) du on [-1,1]

e² - 1

Find f for which f''(x) = 2x + 1, f(0)= 1, f'(0)= 3

f(x) = (1/3)x³ + ½x² + 3x +1

Determine a region whose area is equal to the given limit, do not evaluate the limit. lim n→∞ (i=1) ∑ (π/7n) tan (iπ/7n)

y = tan(x) on [0, (π/7)]

Determine a region whose area is equal to the given limit, do not evaluate the limit. lim n→∞ (i=1) ∑ (7 + (2i/n))¹² (2/n)

y= (7 +x)¹² on [0, 2]

Evaluate the integral ∫ 2/√(1 - t²) dt on [0, √(2)/2]

π/2

Find the most general derivative of f(x)= -1/(³√x) + sec²x

∫ -(3/2)x^(2/3) + tan x + C

Find the most general antiderivative of the function. g(x) = (6 - 7x³ + 3x⁶)/x⁶

G(x) = -(6/5)x⁻⁵ + (7/2)x⁻² + 3x + C

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. ∫ √(x² + 5) dx, n = 4 on [2, 10]

M₄ = 51.6819