Integration using substitution

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Integration using substitution

The chain rule was used to turn complicated functions into simple functions that could be differentiated. This was done using a substitution. The same is true for integration. Each of the following integrals can be simplified using a substitution...

Examples

To integrate by substitution we have to change every item in the function from an 'x' into a 'u', as follows.

 

  1. Choose your substitution u = f(x)
  2. Replace the dx
  3. Change the limits
  4. Now integrate with respect to u

 

Example 1:

To find:

Example 1

we use the substitution u = (3x − 2)

Differentiate this to get

Example 1

Or,

Example 1

This turns the function into:

Example 1

Example 2:

To find:

Example 2

we use the substitution u = x2 − 1.

Differentiate this to get

Example 2

Or,

Example 2

This turns the function into:

Example 2

Example 3:

To find:

Example 3

we use the substitution u = sin x.

Differentiate this to get

Example 3

Or, cos x dx = du

We also need to change the limits.

When x = 0, u = sin 0 = 0,

and when x = π/2, u = sin π/2 = 1.

These changes turn the integration into:

Example 3

Once you are familiar with using substitution it is possible to see what the answer will be without having to go through the stages of actually using the substitution.

This understanding comes with a familiarity with differentiation by inspection and uses the quick differentiation rules (see the differentiation Learn-It for these rules).

These give us the following quick formulae for integration that are based upon calculating where the derivative came from.

Each of these rules can be found by using the substitution u = f(x).

Rules

When integrating cosnx and sinnx there are two identities that are useful.

If the power, n, is odd use the identity cos2x + sin2x = 1

If the power, n, is even use the fact that: cos2x = 2cos2x − 1 = 1 − 2sin2x.

This gives the 2 substitutions:

Substitutions

There are some integrals that need a substitution but are not 'perfect integrals' - they cannot be integrated by inspection.

For these we simply follow through the normal procedure and rewrite the function, the dx, and the limits, in terms of u.

Example:

Find:

Example

using the substitution u = x - 4.

If u = x − 4, then du/dx = 1, and du = dx.

(Replace with √(x − 4) with √u, and dx with du.)

Rearrange u = x − 4, to get x = u + 4.

(This replaces the 'x' with u + 4)

When x = 4, u = 0; when x = 8, u = 4.

(This changes the limits.)

Put these together to get:

Example

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