Calculus Of One Real Variable –
By Pheng Kim Ving |
10.4 |
An integer has 2 possibilities for parity: even or odd. So a
set of 2 integers say m
and n have 2 x
2 = 4 possibilities for
parity, as follows:
“m or n is odd” means “either m or n is odd or both are odd”.
Calculate:
Solution
EOS
An integral of the form:
We've got the integral of a polynomial in u, which can readily be found.
Don't forget to return to the original variable x.
Similarly, if n is
odd, then the substitution u
= sin x
can be utilized.
Both m And n Are Even
Compute:
An integral of the form:
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m Is Even And n Is Either Even Or Odd
Evaluate:
Solution
EOS
An integral of the form:
We've obtained the integral of a polynomial in u, which can readily be done. Don't forget to return to the original variable x.
Both m And n Are Odd
Find:
Solution
EOS
An integral of the form:
We've got the integral of a polynomial in u, which can handily be computed. Then we return to the original variable x.
m Is Odd And n Is Even
If we extract sec^{2} x from sec^{m}
x to form sec^{2} x dx
because the derivative of tan x is sec^{2}
x, then we would have to make
the substitution u = tan x,
so sec^{m}^{–2} x would have to be changed to
an expression involving only integer powers of tan x
and constants using the identity 1 + tan^{2} x = sec^{2}
x, which is impossible because
m – 2 is odd.
If we extract sec x tan x
from sec^{m}
x tan^{n} x
to form sec x tan
x dx
because the derivative of sec x is sec x tan x, then
we would have to make the substitution u
= sec x,
so tan^{n}^{–1} x would have to be changed to
an expression involving only
integer powers of sec x and constants using the
identity 1 + tan^{2} x = sec^{2}
x, which is impossible because
n – 1 is odd.
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Calculate:
Solution
EOS
Integrals of the form:
and the substitution either u = cot x or u = csc x.
Problems & Solutions |
Note
So they differ only by a constant. They're actually equivalent
up to different choices of the constant of integration. Recall
that indefinite integrals are general antiderivatives. If both F_{1}(x) and F_{2}(x) are particular
antiderivatives of f(x), then we
can employ either F_{1}(x) + C or F_{2}(x) + C as the general antiderivative of f(x). Any 2 antiderivatives of a function differ
from each other by a constant; see Section
5.7 Part 2. Thus F_{1}(x) and F_{2}(x)
differ by a constant, and so do F_{1}(x) + C
and F_{2}(x) + C.
If your answer looks different from the one provided, then just differentiate it to see if it's also a correct one.
1. Calculate:
Solution
2. Compute:
Solution
Let v = tan u. Then dv = sec^{2} u du. So:
3. Evaluate:
Solution
4. Find:
Solution
5. Calculate:
Solution
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