The Chain Rule and Integration by Substitution

The Chain Rule and Integration by Substitution

Recall: The chain rule for derivatives allows us to

differentiate a composition of functions:

derivative

[ f (g(x))]' = f '(g(x))g'(x)

antiderivative

The Chain Rule and Integration by Substitution

Suppose we have an integral of the form

f (g(x))g'(x)dx where F'= f .

composition of functions

derivative of Inside function

F is an antiderivative of f

Then, by reversing the chain rule for derivatives,

we have f (g(x))g'(x)dx = F(g(x)) + C.

integrand is the result of differentiating a composition

of functions

Example

Integrate

2x + 5 dx x2 + 5x - 7

Integration by Substitution

Algorithm:

1. Let u = g(x) where g(x) is the part causing problems and g'(x) cancels the remaining x terms in the integrand.

2. Substitute u = g(x) and du = g'(x)dx into the

integral to obtain an equivalent (easier!)

integral all in terms of u.

f (g(x))g'(x)dx = f (u)du

Integration by Substitution

Algorithm: 3. Integrate with respect to u, if possible.

f (u)du = F(u) + C

4. Write final answer in terms of x again.

F(u) + C = F(g(x)) + C

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