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
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
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
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
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
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)
Use the differential equation and the specified initial condition to find y. dy/dx=1 / √(36 − x²), y(0) = π
arcsin(x/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
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))
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
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
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)
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 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
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 area of the region bounded by y = 11 / (36 − x2) and y = 1.
10 - (11/6)ln(11)
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)
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
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)
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
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