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University of Ottawa - Mobius A8 MAT1320
Calculus I (University of Ottawa)
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Question 1: (1 point)
The goal of this question is to guide you through the evaluation of the following integral, using an appropriate trigonometric substitution:
x3
∫ −−−−−− dx
√x2 + 16
a) First, select one of the following trig identities to be the backbone of your substitution:
sec2 (θ) − 1 = tan2 (θ)
[ ]
tan2 (θ) + 1 = sec2 (θ)
[ ]
1 − sin2 (θ) = cos2 (θ)
[ ]
Hint: Choose the trigonometric identity whose left side most resembles x2 + 16 . The right side of each identity is a perfect square
which will ultimately help you to eliminate the radical and simplify the integral!
−−−−−− −−−−−−
b) Next, factor out 16 from the radical to obtain an expression of the form √x2 + 16 = 4√A2 + 1 .
What is A ?
A = __________ .
FORMATTING: The expression for A will depend on x. Since we cannot type the Greek letter θ (“theta”) in Möbius answers, we will
use the variable Q in place of the traditional θ , for the remaining parts of this question.
c) Choose a substitution for x so that the expression A2 + 1 (in terms of x) is exactly the left side of the trig identity from a).
(a) x = 1 sin(Q) (where − π ≤ Q ≤ π )
4 2 2
(b) x = 4 sin(Q) (where − π ≤ Q ≤ π )
2 2
(c) x = 4 sec2 (Q) (where 0 ≤ Q < π or π < Q ≤ π)
2 2
(d) x = 4 sin2 (Q) (where − π ≤ Q ≤ π )
2 2
(e) x = 1 sec(Q) (where 0 ≤ Q < π or π < Q ≤ π)
4 2 2
(f) x = 4 sec(Q) (where 0 ≤ Q < π or π < Q ≤ π)
2 2
(g) x = 4 tan(Q) (where − π < Q < π )
2 2
(h) x = 4 tan2 (Q) (where − π < Q < π )
2 2
(i) x = 1 tan(Q) (where − π < Q < π )
4 2 2
d) Using the substitution from c), write Q in terms of x using the appropriate inverse trig function.
FORMATTING: Pay attention to Möbius syntax for inverse trig functions! e.g. sin−1 (a) is written as arcsin(a), sec−1 (a) is written
as arcsec(a), and tan−1 (a) is written as arctan(a), etc.
Downloaded by olinder seth ()
University of Ottawa - Mobius A8 MAT1320
Calculus I (University of Ottawa)
Scan to open on Studocu
Studocu is not sponsored or endorsed by any college or university
Downloaded by olinder seth ()
, lOMoARcPSD|59658805
Question 1: (1 point)
The goal of this question is to guide you through the evaluation of the following integral, using an appropriate trigonometric substitution:
x3
∫ −−−−−− dx
√x2 + 16
a) First, select one of the following trig identities to be the backbone of your substitution:
sec2 (θ) − 1 = tan2 (θ)
[ ]
tan2 (θ) + 1 = sec2 (θ)
[ ]
1 − sin2 (θ) = cos2 (θ)
[ ]
Hint: Choose the trigonometric identity whose left side most resembles x2 + 16 . The right side of each identity is a perfect square
which will ultimately help you to eliminate the radical and simplify the integral!
−−−−−− −−−−−−
b) Next, factor out 16 from the radical to obtain an expression of the form √x2 + 16 = 4√A2 + 1 .
What is A ?
A = __________ .
FORMATTING: The expression for A will depend on x. Since we cannot type the Greek letter θ (“theta”) in Möbius answers, we will
use the variable Q in place of the traditional θ , for the remaining parts of this question.
c) Choose a substitution for x so that the expression A2 + 1 (in terms of x) is exactly the left side of the trig identity from a).
(a) x = 1 sin(Q) (where − π ≤ Q ≤ π )
4 2 2
(b) x = 4 sin(Q) (where − π ≤ Q ≤ π )
2 2
(c) x = 4 sec2 (Q) (where 0 ≤ Q < π or π < Q ≤ π)
2 2
(d) x = 4 sin2 (Q) (where − π ≤ Q ≤ π )
2 2
(e) x = 1 sec(Q) (where 0 ≤ Q < π or π < Q ≤ π)
4 2 2
(f) x = 4 sec(Q) (where 0 ≤ Q < π or π < Q ≤ π)
2 2
(g) x = 4 tan(Q) (where − π < Q < π )
2 2
(h) x = 4 tan2 (Q) (where − π < Q < π )
2 2
(i) x = 1 tan(Q) (where − π < Q < π )
4 2 2
d) Using the substitution from c), write Q in terms of x using the appropriate inverse trig function.
FORMATTING: Pay attention to Möbius syntax for inverse trig functions! e.g. sin−1 (a) is written as arcsin(a), sec−1 (a) is written
as arcsec(a), and tan−1 (a) is written as arctan(a), etc.
Downloaded by olinder seth ()
