## Solve the following integral using u = tan(x), ∫

MAT 136: Calculus I – Spring 2020

MAT 136: Calculus I – Spring 2020 Exam 4

Instructions: Answer each of the following questions completely. To receive full credit, you must show sufficient work for each of your answers (unless stated otherwise). How you reach your answer is more important than the answer itself.

1.a Solve the following integral using u = tan(x), ∫

2 tan(x) sec2(x)dx. (2.5 points)

1.b Solve the following integral using u = sec(x), ∫

2 tan(x) sec2(x)dx. (2.5 points)

1.c What is the relationship between tan2(x) and sec2(x) that allows both 1.a and 1.b to be true? (5 points)

2 Find the local maximum(s) of f(x) = ∫ x −20(t− 5)

3(t+ 3)2(t− 2)6(t+ 7)9(t− π)dt. (10 points)

1

MAT 136: Calculus I – Spring 2020

3 A race car driver is driving laps around a track. Every 2 hours the speed of the car is being noted in mph. The speed of the car at each of these 2 hour intervals is given in the table below. Use the table below to estimate the total distance traveled over the 20 hours of driving. (10 points)

t=0 t=2 t=4 t=6 t=8 t=10 t=12 t=14 t=16 t=18 t=20

0 50 70 90 100 110 130 100 90 50 20

4 Let f(x) be an integrable function with the following properties,

∫ 5 0 f(x)dx = 10

∫ 9 3 f(x)dx = 5

∫ 5 3 f(x)dx = −2.

What is ∫ 0 9 f(x)dx? Make sure to show every step. (7 points)

2

MAT 136: Calculus I – Spring 2020

5 Solve the following integrals. (5 points each)

a. ∫ 2 0 a

2×3 + bx2 + c3dx b. ∫ 2 0 a

2×3 + bx2 + c3dc

c. ∫

(12x+ 5)f ′(6×2 + 5x+ 2)dx d. ∫

2 sin(2x+ 5)dx

6 Let f(x) be depicted via the graph below. Every shape is either a half circle or a straight line. (5 points each)

a. ∫ 10 0 f(x)dx

b. ∫ 2 0 f(2x+ 5)dx

c. Find L5 on the interval [5, 10].

3

MAT 136: Calculus I – Spring 2020

7 Let f(x) = √

sin(x) + x3. Find d

dx

∫ cos(x) 0

f(x)dx. (15 points)

8 Mark if the following statements are true or false, no justification is needed. (3 points each)

a. If a is a real number then ∫

sina(x) cos(x)dx = sina+1(x).

b. If a and b are constansts then d

dx

∫ b a f(x)dx = 0.

c. If f(x) = g(x) then ∫ f(x)dx =

∫ g(x)dx.

d. If f(x) = g(x) for all x on the interval (a, b) then ∫ b a f(x)dx =

∫ b a g(x)dx.

4

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