10.6 Other Substitutions

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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 = x^1/6. 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 x^1/2 and x^1/3, then let x = u⁶ or u = x^1/6. So x^1/2 = (u⁶)^1/2 = u³, x^1/3 =
(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 = u^p, where p is a common multiple
of all the denominators n₁, n₂, …, 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 n₁, n₂, …, 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:

Example 3.1

Evaluate:

Solution

EOS

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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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