Calc 2
Find the indefinite integral. (Use C for the constant of integration.) ∫19(tan(x))(ln(cos(x))dx
(-19/2)ln(cos(x))² +C
Find the value of c that makes the function continuous at x = 0. f(x) = 8x − 4 sin 2x / 5x3, x ≠ 0 c, x = 0
16 / 15
Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis. √ (9 − x²)
18π
Find the area of the given region. Use a graphing utility to verify your result. (Round your answer to two decimal places.)
2.08
The volume of the torus shown in the figure is given by the integral below, where R > r > 0. Find the volume of the torus. 8πR[∫ from 0 to r (√(r²-y²)) dy
2π²r²R
Find the derivative of the function. y = 5 ln(tanh x/ 2)
5csch(x)
Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis. y = 4x - x² x = 0 y = 4
8π/3
Find the indefinite integral. (Use C for the constant of integration.) ∫t⁷ (8√(t⁸ - 74)) dt
C + (1/9)(t⁸ - 74)^9/8
Find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines. y = √(x) y = 0 x = 6 a) the x-axis b) the y-axis c) the line x=6 d) the line x = 9
a) 18π b) 144/5 (√(6)*π) c) 96/5 (√(6)*π) d) 216/5 (√(6)*π)
Determine whether the improper integral converges or diverges. ∫ from 0 to ∞ (x¹²e^-x) dx
converges
Determine whether the improper integral converges or diverges. ∫ from 4 to 8 (1 / √(64 - x^2) dx
converges
Determine whether the improper integral diverges or converges. ∫ from 2 to 5 ( 1 / √(x²-4)) dx
converges
Find the derivative of the function. y = 5/2(1/2ln((x+1)/(x-1)) + arctanx)
dy/dx = 5 / 1 - x⁴
Use the shell method to find the volume of the solid generated by revolving the plane region about the line x = 5. y = √x y = 0 x = 4
(416/15)π
Find the limit. lim x→−∞ sinh(x)
-∞
Find the area of the region bounded by y = 11 / (36 − x2) and y = 1.
10 - (11/6)ln(11)
Consider the following. lim x→∞ (1 + 1/x)^x (a) Describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L'Hôpital's Rule if necessary. (c) Use a graphing utility to graph the function and verify the result obtained in part (b)
a. 1^∞ b. e
Use implicit differentiation to find an equation of the tangent line to the graph of the equation at the given point. x² + x arctan y = y − 1, (-π/4, 1)
(-2π / 8+π)x+1 - (π² / 16 + 2π)
Evaluate the definite integral. ∫from ln8 to ln20 (4e^-x / (√(1 - 16e^-2x)) dx
(π/6) - sin^-1(1/5)
Write the expression in algebraic form. [Hint: Sketch a right triangle.] cos(arcsin((x - h) / r))
(√(r² - (x - h) ²)) / r
Consider the following functions. f(x) = 3^x g(x) = 2x + 1 Sketch the region bounded by the graphs of the functions. Find the area of the region. (Round your answer to three decimal places.)
0.180
Find or evaluate the integral by completing the square. (Round your answer to three decimal places.) ∫ from 0 to 1 ( dx / x² − 2x + 2)
0.785
Find the arc length of the graph of the function over the indicated interval. (Round your answer to three decimal places.) y = ln(cos(x)), [0, π/4]
0.881
Find the area between the graph of y = sin x and the line segment joining the points (0, 0) and (7π/6, −1/2), as shown in the figure. (Round your answer to four decimal places.)
1 + √(3)/2 + 7π/24
Use the value of the given hyperbolic function to find the values of the other hyperbolic functions at x. tanh(x) = 1/2 sinh(x) = csch(x) = cosh(x) = sech(x)= coth(x)=
1. √3 / 3 2. √3 3. 2√3 / 3 4. √3 / 2 5. 2
Consider the following functions. f(x) = 3√(x − 7) g(x) = x − 7 Sketch the region bounded by the graphs of the functions. Find the area of the region.
1/2
Evaluate the limit, using L'Hôpital's Rule if necessary. (If you need to use or -, enter INFINITY or -INFINITY, respectively.) lim x→0 arctan(x) / sin(6x)
1/6
Find the indefinite integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) ∫ x² / x-4 dx
16ln(|x-4|) + C + x²/2 +4x
Find the arc length of the curve over the given interval. (Round your answer to three decimal places.) y = ln x, [1, 3]
2.302
Find the indefinite integral. (Use C for the constant of integration.) ∫ (ln x)⁵ / x dx
C + (ln⁶(x) / 6)
Find the indefinite integral. (Use C for the constant of integration.) ∫ (t / t⁴ +81) dt
C + 1/18tan^-1 (t²/9)
Find the indefinite integral. (Use C for the constant of integration.) ∫e^x √(1 - e^2x) dx
C + 1/2(e^x √(1-e^2x) +sin^-1 (e^x))
Find the integral. (Use C for the constant of integration.) ∫sin(−6x)cos(5x) dx
C + 1/22(11cos(x) +cos(11x))
Find the integral. (Note: Solve by the simplest method - not all require integration by parts. Use C for the constant of integration.) ∫x⁷ln(x) dx
C + 1/64 x⁸(8ln(x) - 1)
Find the integral. (Note: Solve by the simplest method - not all require integration by parts. Use C for the constant of integration.) ∫e^4x sin(x) dx
C +1/17 e^4x(4sin(x) - cos(x))
Find the integral involving secant and tangent. (Use C for the constant of integration.) ∫sec⁶6x dx
C +1/6 (tan(6x) + 2/3 tan(6x)³ + 1/5 tan(6x)⁵
Find the indefinite integral. (Use C for the constant of integration.) ∫(x+3) / (√(4 - (x - 2)²) dx
C - √(4x - x²) - 5sin^-1 (1 - x/2)
Use the differential equation and the specified initial condition to find y. dy/dx=1 / √(36 − x²), y(0) = π
arcsin(x/6) + π
Consider the following. f(x) = a^x − 1 / a^x + 1 for a > 0, a ≠ 1 Show that f has an inverse function. Then find f ^-1.
f^-1(x) = (1 / ln(a)) (ln(x+1 / 1-x))
Find the derivative of the function. (Hint: In some exercises, you may find it helpful to apply logarithmic properties before differentiating.) h(x) = log₂((x√(x-9))/ 2)
h'(x) = (3x-18) / 2ln(2)x(x-9)
Evaluate the definite integral. Use a graphing utility to verify your result. ∫from 0 to 7 (8 / 9x² +10x +1) dx
ln(8)
Let f(t) be a function defined for all positive values of t. The Laplace Transform of f(t) is defined by the following integral, if the improper integral exists. F(s) = ∫ from 0 to ∞ ( e^−st f(t) ) dt Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. f(t) = cos at
s / s² + a²
Solve the differential equation. (Use C for the constant of integration.) dy/dx = 1 / x√(100x² - 1)
sec^-1 (10|x|) +C
Use a graphing utility to graph the function. f(x) = arccos(x/3)
x=3
Use implicit differentiation to find an equation of the tangent line to the graph at the given point. x + y − 1 = ln(x⁸ + y⁷), (1, 0)
y = 7x - 7
Find an equation of the tangent line to the graph of the function at the given point. y = x²e^x − 2xe^x + 2e^x, (1, e)
y=ex
Find the area of the unbounded shaded region. y = 1/ x²+1
π
Evaluate the definite integral. ∫ of 0 to √(3) / 5 (1 / 1+25x²) dx
π/15
Evaluate the integral. ∫ from 0 to √(2)/4 (2 / √(1 - 4x²) dx
π/4
Find the area of the region. y = 2e^x / 1 + e^2x
π/6
