Calculus Of One Real Variable –
By Pheng Kim Ving

10.6 
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Go To Problems
& Solutions
1. Completing The Square 
Calculate:
Integrals whose
integrands involve the quadratic expression ax^{2} + bx + c but aren't polynomials can often be handled
as
follows. First, complete the square:
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2. Elimination Of All Fractional Exponents 
Compute:
Let x = u^{6} or u = x^{1/6}. Then dx = 6u^{5} du. So:
EOS
If an integral contains 2 or more fractional exponents, a
substitution can be used to simultaneously eliminate all of them
at once. For example, if the integrand contains x^{1/2}
and x^{1/3}, then let x = u^{6} or u
=
x^{1/6}. So x^{1/2} = (u^{6})^{1/2} = u^{3}, x^{1/3} =
(u^{6})^{1/3} = u^{2}, and dx
=
6u^{5} du.
In general suppose we have:
and we want to eliminate all the fractional exponents. Then
we have to substitute x
= u^{p},
where p is a
common multiple
of all the denominators n_{1}, n_{2}, ..., n_{k}. Thus we choose the
simplest common multiple, which is the LCM (least common
multiple). Consequently p
is the LCM of all the denominators n_{1}, n_{2}, ..., n_{k}. When substituting x = u^{p} don't forget to derive
that u = x^{1/}^{p}.
Hence the fractional exponents are eliminated and an
integral in u
with integer exponents is obtained. If the integrand is a
fraction where the degree of the numerator exceeds or equals that of the
denominator, perform long division. Next, apply
integration formulas and/or utilize techniques presented in earlier sections.
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3. Rational Functions Of sine And/Or cosine 
The integrand in this integral:
Evaluate:
EOS
Problems & Solutions 
1. Calculate:
Solution
2. Compute the definite integral:
Solution
3. Evaluate:
Solution
4. Find:
Solution
5. Calculate:
Solution
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