Calculus Of One Real Variable By Pheng Kim Ving
Chapter 10: Techniques Of Integration Section 10.6: Other Substitutions

 

10.6
Other Substitutions

 

 

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1. Completing The Square

 

Example 1.1

 

Calculate:

 

 

Solution

EOS

 

Integrals whose integrands involve the quadratic expression ax2 + 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

 

Example 2.1

 

Compute:

 

 

Solution

Let x = u6 or u = x1/6. Then dx = 6u5 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 x1/2 and x1/3, then let x = u6 or u = x1/6. So x1/2 = (u6)1/2 = u3, x1/3 =
(u6)1/3 = u2, and dx = 6u5 du.

 

In general suppose we have:

 

 

and we want to eliminate all the fractional exponents. Then we have to substitute x = up, where p is a common multiple
of all the denominators n1, n2, ..., nk. Thus we choose the simplest common multiple, which is the LCM (least common
multiple). Consequently p is the LCM of all the denominators n1, n2, ..., nk. When substituting x = up don't forget to derive
that u = x1/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:

 

 

 

 

Example 3.1

 

Evaluate:

 

 

Solution


EOS

 

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Problems & Solutions

 

1. Calculate:

 

 

Solution

 

 

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2. Compute the definite integral:

 

 

Solution

 

 

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3. Evaluate:

 

 

Solution

 

 

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4. Find:

 

 

Solution

 

 

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5. Calculate:

 

 

Solution

 

 

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