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Unformatted text preview: Math 116 - Lab 9 - Fall 2011.
Integration by Substitution, Average Value of a Function, Curve Length (Sections 6.1, 7.7
and 7.1 of L/G).
You are to provide full solutions to the following problems. You are allowed to collaborate with your
classmates, use your notes and textbook and ask the TA for guidance. Direct copying of solutions
is not encouraged, nor is it allowed or ethical. First name: Last name:
Student number: 1 Math 116 - Lab 9 - Fall 2011. Student number: 1. Compute the following integrals by substitution.
l) sin(3x) dx
exp(t + 3) dt
(3t3 + 7)4 t2 dt
dx (Hint: recall the derivative of arctan(x).)
1 + x6
dt (Hint: recall the derivative of arctan(x) and complete the square.)
t2 + 2t + 5
t exp(t2 + 1)
dt (Hint: recall the derivative of arctan(x).)
1 + exp(2t2 + 2)
2 (t2 + 1)
sin2 (t) cos(t) dt
sin5 (t) cos10 (t) dt
2 2t exp(t2 ) dt m)
(t + 1)2 −1 o) 3 cosh(2x + 1) dx 0
dt (Hint: recall the derivative of arcsin(x).)
1 − t4 2. Compute 2 sin(x) cos(x) dx in three diﬀerent ways: use the substitution u = sin(x),
use the substitution u = cos(x), and use the trigonometric identity sin(2x) = 2 sin(x) cos(x).
Explain why the three results you obtain are essentially the same result.
3. a) Determine the average value of f (x) = x + 7 over [1, 2] in two diﬀerent ways (by
integration and graphically).
b) Determine the average value of f (t) = ln(t)/t over [1, 3]. 4. a) Determine the curve length of f (x) = x3/2 between x = 0 and x = 1.
b) Determine the curve length of f (x) = cosh(x) between x = − ln(2) and x = ln(2).
(Hint: the integrand simpliﬁes considerably. . . ) 2 ...
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