Sometimes an unfamiliar-looking integral can be made into a familiar one by substituting a new variable for an old one. For example, we know how to integrate — the answer is — but what about
Let . Differentiating both sides, we have , or ,so
This technique is known as a change of variable or a substitution. (Because the letter u is of- ten employed, you may also see it called -substitution.)
In the case of a definite integral, we have to remember to change the limits of integration to reflect the new variable.
Example
Evaluate .
As before, let .
Here the notation means to evaluate the function at 7 and 9, and sub- tract the former from the latter. The result is
Sometimes, as in the next example, a clever substitution is the secret to doing a seemingly impossible integral.
Example
Evaluate
The only hope for reducing this to a form we can do is to let . Then , so
Example really isn't so tricky, since there was only one logical choice for the substitution that had any hope of working. The following is a little more dastardly.
Example
Evaluate
The substitution that works is . First let’s see what this does to the expression . The familiar identity
when divided by , gives
so becomes . But differentiating both sides of gives
so the integral becomes
Integrate(x) 1/(1+x^2) ArcTan(x)
Another possible answer is that you can usually smell the possibility of this type of substitution, involving a trig function, when the thing to be integrated contains something reminiscent
of the Pythagorean theorem, as suggested by Figure 5.2. The looks like what
you’d get if you had a right triangle with legs 1 and , and were
using the Pythagorean theorem to find its hypotenuse.
Example
Evaluate
The looks like what you’d get if you had a right triangle with hypotenuse 1 and a leg of length , and were using the Pythagorean theorem to find the other leg, as in Figure 5.3. This motivates us to try the substitution , which gives and . The result is
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