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The Integral
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6.2
Substitution
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3. Substitution
The chain rule for derivatives gives us an extremely useful technique for finding antiderivatives. This technique is called change of variables or substitution.
We know that to differentiate a function like (x2 + 1)6, we first think of the function as g(u) where u = x2 + 1 and g(u) = u6. We then compute the derivative, using the chain rule, as

4. Substitution
Any rule for derivatives can be turned into a technique for finding antiderivatives by writing it in integral form. The integral form of the formula is
But, if we write g(u) + C = ∫ g(u) du, we get the following interesting equation:
This equation is the one usually called the change of variables formula.

5. Substitution
We can turn it into a more useful integration technique as follows.
Let f = g(u)(du/dx). We can rewrite the change of variables formula using f :
In essence, we are making the formal substitution

6. Substitution Substitution Rule
If u is a function of x, then we can use the following formula to evaluate an integral:

7. Substitution
Rather than use the formula directly, we use the following step-by-step procedure:
1. Write u as a function of x.
2. Take the derivative du/dx and solve for the quantity dx in
terms of du.
3. Use the expression you obtain in step 2 to substitute for
dx in the given integral and substitute u for its defining
expression.

8. Example 1 – Substitution
Find ∫ 4x(x2 + 1)6 dx.
Solution:
To use substitution we need to choose an expression to be u.
There is no hard and fast rule, but here is one hint that often works:
Take u to be an expression that is being raised to a power.
In this case, let’s set u = x2 + 1.

9. Example 1 – Solution
cont’d
Continuing the procedure above, we place the calculations for step 2 in a box.
Write u as a function of x.
Take the derivative of u with respect to x.
Solve for dx: dx = du.

10. Example 1 – Solution ∫ 4x(x2 + 1)6 dx = ∫ 4xu6 du
cont’d
Now we substitute u for its defining expression and substitute for dx in the original integral:
∫ 4x(x2 + 1)6 dx = ∫ 4xu6 du
= ∫ 2u6 du.
We have boiled the given integral down to the much simpler integral ∫ 2u6 du, and we can now write down the solution:
Substitute for u and dx.
Cancel the xs and simplify.
Substitute (x2 + 1) for u in
the answer.

11. Shortcuts

12. Shortcuts Shortcuts: Integrals of Expressions Involving (ax + b)
Rule Quick Example

13. Shortcuts
cont’d
Rule Quick Example

14. Shortcuts (Optional) Mike’s Shortcut Rule: Integrals of More
General Expressions
Rule Quick Example

15. Shortcuts
cont’d
Rule Quick Example