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Problem 4.9.21
Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.
g(s) = 1 / (s² + 1)
Problem 4.9.23
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (3x⁵ - 5x⁹) dx
Problem 4.9.25
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (4√x - (4 /√x)) dx
Problem 4.9.27
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (5s + 3)² ds
Problem 4.9.29
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (3x ¹⸍³ + 4x ⁻¹⸍³ + 6) dx
Problem 4.9.31
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (3x + 1) (4 - x) dx
Problem 4.9.33
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (3/x⁴ + 2 - 3/x²)
Problem 4.9.35
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((4x⁴ - 6x²) / x ) dx
Problem 4.9.37
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (x² - 36) / (x - 6) dx
Problem 4.9.39
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (csc² Θ + 2Θ² - 3Θ) dΘ
Problem 4.9.41
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((2 + 3 cos y)/sin² y)dy
Problem 4.9.43
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (sec² x - 1) dx
Problem 4.9.45
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (sec² Θ + sec Θ tan Θ)dΘ
Problem 4.9.51
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (1/2y)dy
Problem 4.9.53
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (6/√(4 - 4x²))dx
Problem 4.9.49
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ sec Θ(tan Θ + sec Θ + cos Θ)dΘ
Problem 4.9.57
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (1/x√(36x² - 36))dx
Problem 4.9.55
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (4/x√(x² - 1))dx
Problem 4.9.61
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (eˣ⁺²) dx
Problem 4.9.63
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((e²ʷ - 5eʷ + 4)/(eʷ - 1))dw
Problem 4.9.65
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((1 + √x)/x)dx
Problem 4.9.67
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (√x(2x⁶ - 4³√)dx
Problem 4.9.109a
107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .
a. Find the velocity of the object for all relevant times.
A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.
Problem 4.9.109d
107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .
d. Find the time when the object strikes the ground.
A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.
Problem 4.9.109b
107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .
b. Find the position of the object for all relevant times.
A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.
Problem 4.9.109c
107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .
c. Find the time when the object reaches its highest point. What is the height?
A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.
Problem 4.9.91
Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.
v(t) = 2t + 4; s(0) = 0
Problem 4.9.93
Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.
v(t) = 2√t; s(0) = 1
Problem 4.9.79
Solving initial value problems Find the solution of the following initial value problems.
g'(x) = 7x(x⁶ - 1/7); g(1) = 2
Problem 4.9.85
Solving initial value problems Find the solution of the following initial value problems.
y'(Θ) = ((√2 cos³ Θ + 1)/cos² Θ); y (π/4) = 3, -π/2 < Θ < π/2