Calculus Help
Name: Integration Formulas Section:
5.4 Integration Formulas Vocabulary Examples
Net Change Theorem The value of a changing quantity equals
plus the of the of
.
F(b) =
or∫ b a F
′(x) =
1. No problem designed yet.
2. ∫ 5 0 (2x
2 − 3x)d x
3. ∫ π 0 (cos x)d x
4. ∫ π/4 −π/4(sin x)d x
5. ∫ 4 −2(x
2 − 2x + 1)d x
6. ∫ 3 2 (x + 2)
3d x
7. ∫ 2 1
x+x5 2x3
d x
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Name: Integration Formulas Section:
8. Given a velocity function v(t) = 3t − 5 (in meters per second) for a particle in motion from time t = 0 to time t = 3, find the net displacement of the particle.
9. Determine the total distance traveled by the particle in #8 (above) over the interval of time [0, 3]
10. If the motor on a motorboat is started at t = 0 and the boat consumes gasoline at a rate of 5 − t3 gal/hr, how much gasoline is used in the first two hours?
11. The active electrical usage of a building is given by the function p(t) = 0.3t3 + 0.1t2 + 0.42 in kilowatts from time t = 2 hours to t = 6 hrs. Determine the how much total energy (in kilowatt-hours)
are used during that time period.
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Name: Substitution Section:
5.5 Substitution Vocabulary Examples
Substitution (with indefinite integrals)
Let u = g(x), where g′(x) is continuous over an interval.
Let f (x) be continuous over the corresponding range of g
and let F(x) be an antiderivative of f (x), then∫ f [g(x)]g′(x)d x =
Substitution (with definite integrals)
Let u = g(x), where g′(x) is continuous over [a, b]. Let
f (x) be continuous over the corresponding range of g, then∫ f [g(x)]g′(x)d x =
Determine the following indefinite integrals using the given u.
1. ∫
x √
x + 1d x; u = x + 1 2. ∫ −
2 sin(π+ √
x) √
x d x; u = π +
√ x3.
∫ x2 √
x−1 d x; u = x − 1
Determine the following indefinite integrals using u substitution. (Be sure to clearly identify u and du.)
4. ∫
(x − 1)4d x 5. ∫
(x − 1)(x2 − 2x)3d x 6. ∫
(3x − 2)−11d x
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Name: Substitution Section:
7. ∫
cos3 θdθ 8. ∫
x(1 − x)99d x 9. ∫
(3x − 1)e3x 2−2xd x
Evaluate the following definite integrals.
10. ∫ 1 0
x2 (x3−3)2
d x 11. ∫ π/4 0
sin θ cos4 θ
dθ 12. ∫ π 0 cos
2(2t) sin(2t)dt
13. ∫ 1/2 0 −2x ln(1 − x
2)d x 14. ∫ 1 0 x √
1 − x2d x 15. ∫ 2 0
t2 √
1+t3 dt
16.
∫ 4√2 1 −
16a3 − 16a(
1 + ( a2 − 1
)2)2 da 17.
∫ 1 0 x √
1 − x2d x 18. ∫ 2 0
t2 √
1+t3 dt
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Name: Fundamental Theorem of Calculus Practice Section:
A.15 Fundamental Theorem of Calculus Practice Use the Fundamental Theorem of Calculus to determine the following derivatives.
1. d d x
∫ x 1
( 3x2 − 4 cos(x4)
) d x 2.
d d x
∫ 7t 0
sin θ cos2 θdθ 3. d d x
∫ 1 √
x
t2
1 + t4 dt
Evaluate the following definite integrals.
4. ∫ 2 −1(x
2 − 3x)d x 5. ∫ 3 −2(x
2 + 3x − 5)d x 6. ∫ 3 −2(t + 2)(t − 3)dt
7. ∫ 1 0 (x
99)d x 8. ∫ 4 1/4
( x2 − 1
x2
) d x 9.
∫ 4 1
1 2 √
x d x
10. ∫ 4 1
2− √
t t2
dt 11. ∫ 2π 0 (8 cos(θ))dθ 12.
∫ −1 −2
( 1 t2 −
1 t3
) dt
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Name: Fundamental Theorem of Calculus Practice Section:
13. The vertical displacement v, in m/s, of a projectile at time t is given by the function v(t) = −9.8t + 84.1.
(a) Determine the average vertical velocity over the interval [0, 3] (b) Determine the average vertical velocity over the interval [0, 9] (c) Determine the average vertical velocity over the interval [9, 10] (d) Determine the average vertical velocity over the interval [0, 18]
14. The velocity v, in m/s, of a particle at time t is given by the function v(t) = 11+x − 3x 2 + x3 + 5.
Determine the average velocity of the particle between t = 0 s and t = 3 s.
Determine the following definite integrals.
15. ∫ 5 −2 |x|d x 16.
∫ 5 0
∣∣∣x2 − 2x − 8∣∣∣ d x 17. ∫ 5 −3
∣∣∣x2 − 2x − 8∣∣∣ d x
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Name: Substitution Practice I Section:
A.16 Substitution Practice I Determine the following indefinite integrals.
1.
∫ ( x √
4x2 + 9 )
d x 2. ∫
(x + 1)4d x
3.
∫ cos3 θdθ 4.
∫ (2x − 3)−7d x
5.
∫ sin3 θdθ 6.
∫ (x2 − 2x)(x3 − 3x2)3d x
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Name: Substitution Practice I Section:
7. Determine the average value of f (x) = x √
25 − x2d x over each of the following intervals.
(a) [0, 5] (b) [−1, 4]
(c) [ −
1 3,
1 3
]
8. Determine the average value of f (x) = 36x 4x2+9
d x over each of the following intervals.
(a) [0, 1.5] (b) [0, 3]
(c) [ −
√ 1155 3 ,
√ 1155 3
]
9. Calculate the area of the shaded region.
f (x) = 2 + cos(πx)
g(x) = 2x − 1
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