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Methods of Integration - Substitution Help

By — McGraw-Hill Professional
Updated on Aug 31, 2011

Introduction to Methods of Integration - Substitution

Sometimes it is convenient to transform a given integral into another one by means of a change of variable. This method is often called “the method of change of variable” or “ u -substitution.”

To see a model situation, imagine an integral

Methods of Integration 7.3 Substitution

If the techniques that we know will not suffice to evaluate the integral, then we might attempt to transform this to another integral by a change of variable x = φ( t ). This entails dx = φ′ ( t ) dt . Also

x = at = φ −1 ( a ) and x = bt = φ −1 ( b ).

Thus the original integral is transformed to

Methods of Integration 7.3 Substitution

It turns out that, with a little notation, we can make this process both convenient and straightforward.

We now illustrate this new paradigm with some examples. We begin with an indefinite integral.

Example 1

Evaluate

∫ [sin x ] 5 · cos x dx .

Solution 1

On looking at the integral, we see that the expression cos x is the derivative of sin x . This observation suggests the substitution sin x = u . Thus cos x dx = du . We must now substitute these expressions into the integral, replacing all x -expressions with u -expressions. When we are through with this process, no x expressions can remain. The result is

u 5 du .

This is of course an easy integral for us. So we have

Methods of Integration 7.3 Substitution

Now the important final step is to resubstitute the x -expressions in place of the u -expressions. The result is then

Methods of Integration 7.3 Substitution

Math Note : Always be sure to check your work. You can differentiate the answer in the last example to recover the integrand, confirming that the integration has been performed correctly.

Example 2

Evaluate the integral

Methods of Integration 7.3 Substitution

Solution 2

We recognize that the expression 2x is the derivative of x2 + 1. This suggests the substitution u = x2 + 1. Thus du = 2 x dx . Also x = 0 ↔ u = 1 and x = 3 ↔ u = 10. The integral is thus transformed to

Methods of Integration 7.3 Substitution

This new integral is a bit easier to understand if we write the square root as a fractional power:

Methods of Integration 7.3 Substitution

You Try It : Evaluate the integral

Methods of Integration 7.3 Substitution

Math Note : Just as with integration by parts, we always have the option of first evaluating the indefinite integral and then evaluating the limits at the very end. The next example illustrates this idea.

Example 3

Evaluate

Methods of Integration 7.3 Substitution

Solution 3

Since cos x is the derivative of sin x , it is natural to attempt the substitution u = sin x . Then du = cos x dx . [Explain why it would be a bad idea to let u = cos x .] We first treat the improper integral. We find that

Methods of Integration 7.3 Substitution

Now we resubstitute the x -expressions to obtain

Methods of Integration 7.3 Substitution

Finally we can evaluate the original definite integral:

Methods of Integration 7.3 Substitution

You Try It : Calculate the integral

Methods of Integration 7.3 Substitution

Find practice problems and solutions for these concepts at: Methods Of Integration Practice Test.

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