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Worksheet 3. Math 231 BD3. Spring 2015. 1/28/2015.
7.2: Trigonometric Integrals
This worksheet is about solving integrals involving trig functions. Remember two fundamental trig equalities:
sin2 x + cos2 x = 1
and
sec2 x − tan2 x = 1
The first goal is to solve integrals of the form:
Z
sinn x cosm x dx
If either n or m are odd, then you have to use substitution:
Z
1. Evaluate sin2 x cos5 x dx.
Z
2. Evaluate
sin3 x cos4 x + sin x cos3 x dx.
Z
3. Use the same techniques to evaluate
cos3 x
√
dx.
sin x
The hardest case is when n and m are both even. In order to solve them we
need more formulas:
4. Use
cos(A ± B) = cos A cos B ∓ sin A sin B
sin(A ± B) = sin A cos B ± cos A sin B
to prove the following trig identities:
sin2 x = 21 (1 − cos 2x)
cos2 x = 12 (1 + cos 2x)
sin x cos x = 12 sin 2x
Now use these three formulas to solve the next two questions:
Z
5. Evaluate sin2 x dx.
Z
6. Evaluate
4 sin2 x cos2 x dx.
The second goal is to solve integrals of the form:
Z
tanm x secn x dx
If m is odd or n is even then use substitution:
Z
7. Evaluate sec x tan3 x dx.
Z
sec4 x tan2 x dx.
8. Evaluate
If m is even and n is odd then there is no easy way to solve it and we will look
at particular examples:
Z
9. Evaluate sec x dx (Hint: Multiply top and bottom of the integral by sec x + tan x)
Z
10. Evaluate
sec3 x dx.
(Hint: Use integration by parts on sec x sec2 x)
Z
11. Use the technique of the last question to evaluate
Z
n
sec x dx in terms of
secn−2 x dx
where n is odd.
Z
12. This also works for other integrals. Find
n
sin x dx in terms of
Z
sinn−2 x dx for n odd.
13. Household electricity is supplied in the form of alternating current that varies from 155
V to -155 V with a frequency of 60 cycles per second (Hertz Hz). The voltage is thus given
by the equation
E(t) = 155 sin(120πt)
where t is the time in seconds. Voltmeters read the RMS (root-mean square) voltage, which
is the square root of the average value of [E(t)]2 over one cycle.
1. Calculate the RMS voltage of household current (you can use previous problems).
2. Many electric appliances require an RMS voltage of 220 V. Find the corresponding
amplitude A needed for the voltage
E(t) = A sin(120πt).