After learning a simple list of antiderivatives, it is time to move on to more complex integrands, which are not at first readily integrable. In these first steps, we notice certain special case integrands which can be easily integrated in a few steps.
Recognizing Derivatives and Reversing Derivative RulesEdit
If we recognize a function as being the derivative of a function , then we can easily express the antiderivative of :
For example, since
we can conclude that
Similarly, since we know is its own derivative,
The power rule for derivatives can be reversed to give us a way to handle integrals of powers of . Since
we can conclude that
or, a little more usefully,
Integration by SubstitutionEdit
For many integrals, a substitution can be used to transform the integrand and make possible the finding of an antiderivative. There are a variety of such substitutions, each depending on the form of the integrand.
The objective of Integration by substitution is to substitute the integrand from an expression with variable x to an expression with variable where
We want to transform the Integral from a function of x to a function of u
|(4)||ie Now equate with|
|(7)||ie We have achieved our desired result|
- Calculate which is and make sure you express the result in terms of the variable u
Integrating with the derivative presentEdit
If a component of the integrand can be viewed as the derivative of another component of the integrand, a substitution can be made to simplify the integrand.
For example, in the integral
we see that is the derivative of . Letting
or, in order to apply it to the integral,
With this we may write
Note that it was not necessary that we had exactly the derivative of in our integrand. It would have been sufficient to have any constant multiple of the derivative.
For instance, to treat the integral
we may let . Then
the right-hand side of which is a factor of our integrand. Thus,
In general, the integral of a power of a function times that function's derivative may be integrated in this way. Since ,
There is a similar rule for definite integrals, but we have to change the endpoints.
Consider the integral
By using the substitution u = x2 + 1, we obtain du = 2x dx and
Note how the lower limit x = 0 was transformed into u = 02 + 1 = 1 and the upper limit x = 2 into u = 22 + 1 = 5.
Proof of the substitution ruleEdit
We will now prove the substitution rule for definite integrals. Let H be an anti derivative of h so
Suppose we have a differentiable function, such that , and numbers and derived from some given numbers, and .
By the Fundamental Theorem of Calculus, we have
Next we define a function by the rule
Then by the Chain rule F is differentiable with derivative
Integrating both sides with respect to and using the Fundamental Theorem of Calculus we get
But by the definition of this equals
which is the substitution rule for definite integrals.
Evaluate the following using a suitable substitution.