Section 7.5

Integrals Involving Roots

When an integral contains a messy root like , a Rationalizing Substitution lets us eliminate the radical entirely by substituting for the root.

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The Rationalizing Substitution

When an integral contains a messy root like , standard u-substitution might fail. A Rationalizing Substitution involves:

  1. Let equal the entire root
  2. Solve for in terms of
  3. Find in terms of
  4. Substitute — the radical is eliminated entirely

The Key Idea

For an integral containing :

Let:

Then:

The scary root function becomes just a polynomial in !

Multiple Roots?

When an integral contains multiple different roots like and , use an exponent that's a common multiple of all indices:

For and , let (LCM of 2 and 3 is 6)

Then and

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Worked Examples

Example 1: Linear Root

Evaluate .

1. Setup the Substitution

Let so that

2. Solve for x and dx
Solve for x:
Find dx:
3. Substitute and Simplify

Expand:

4. Integrate
5. Back-Substitute

Replace :

Example 2: Root in Denominator

Evaluate .

1. Setup the Substitution

Let so that

2. Find dx
3. Substitute
4. Long Division

Since degree of numerator ≥ degree of denominator, perform division:

So the integral becomes:

5. Integrate
6. Back-Substitute

Example 3: Multiple Roots (Different Indices)

Evaluate .

1. Choose Common Exponent

We have and

LCM of 3 and 2 is 6, so let

2. Express Everything in Terms of u
and
3. Substitute and Simplify

Factor and simplify:

4. Polynomial Division

Divide by :

5. Integrate
6. Back-Substitute

Replace :

Or equivalently:

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Practice Quiz

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