Download - Calc 5.2b
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Integrating trigonometric functions
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Ex. 7 p.335 u-Substitution and the Log Rule
We can solve differential equations using the log rule as well.Solve the differential equation 1
ln
dy
dx x x
Solution - separate y things from x things and integrate both sides. Put the “plus C” on right side only.
1
lndy dx
x x 1
lndy dx
x x
1
lny dx
x x
There are three basic choices for u: u = x, u = x ln x, and u = ln x. The first two don’t fit the u’/u pattern. If I rewrite the function to be 1
lnxy dxx
the pattern fits because u = ln x and du = (1/x)dx
Rewrite with u-substitution:'u
y dxu
ln u C Back-substitute: ln lny x C
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Up until now, we didn’t know how to integrate tan x, cot x, sec x, and csc x. With the Log rule, we can now do integration of these functions.
Ex 8 p. 336 Using a trig identity to integrate using log rule
Find tan x dxRewrite with trig identity:
sin
cos
xdx
x
Let u = cos x. Then du = – sin x dx
'udxu
ln cos x C
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Ex 9 p. 336 Derivative of the secant functionThis problem needs a creative step, multiplying and dividing by the same quantity to make it work.
sec x dxFindsec tan
secsec tan
x xx dx
x x
2sec sec tan
sec tan
x x xdx
x x
Let u = sec x + tan x. (the denominator) Then du = sec x tan x + sec2 x, which is the numerator!
'udxu
Integrate and back-substitute
ln u C ln sec tanx x C
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The other two are left as problems in the homework.
I remember these log ones by realizing if they are co- things, they have a negative in front. Secant and tangent go together, as do cosecant and cotangent.
These can be written in different forms, see #83-86
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Ex 10 p. 337 Integrating Trigonometric Functions
Evaluate 22
4
1 cot x dx
Using Pythagorean Identity, 1 + cot2 x = csc2 x
22
4
csc x dx
2
4
csc x dx
2
4ln csc cotx x
ln 1 0 ln 2 1
ln csc cot ln csc cot2 2 4 4
ln 2 1 0.881372587
Be careful with parentheses if graphing and getting definite integral with the calculator.
ln 2 1
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Last but not least, Ex 11 p. 337 Finding an average value
Recall that average value refers to average height, which would be area divided by width. 1
( )b
a
f x dxb a
Find the average value of f(x) = tan x on the interval [0, π/4]4
0
1tan
04
x dx
4ln cos ln cos 04
4 2ln ln 1
2
4 2ln
2
0.441
4
0
4ln cos x
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1
π/4
y = 0.441 is avg value
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5.2b p. 338 29-53 every other odd (skip 45), 65,70, 73, 83, 85