Computing Indefinite Integrals
In the previous section we started looking at indefinite
integrals and in that section we concentrated almost exclusively on notation,
concepts and properties of the indefinite integral. In this section we need to
start thinking about how we actually compute indefinite integrals. We�ll start
off with some of the basic indefinite integrals.
The first integral that we�ll look at is the integral of a
power of x.
The general rule when integrating a power of x we
add one onto the exponent and then divide by the new exponent. It is clear
(hopefully) that we will need to avoid
 in
this formula. If we allow
 in
this formula we will end up with division by zero. We will take care of this
case in a bit.
Next is one of the easier integrals but always seems to
cause problems for people.
If you remember that all we�re asking is what did we
differentiate to get the integrand this is pretty simple, but it does seem to
cause problems on occasion.
Let�s now take a look at the trig functions.
Notice that we only integrated two of the six trig
functions here. The remaining four integrals are really integrals that give the
remaining four trig functions. Also, be careful with signs here. It is easy to
get the signs for derivatives and integrals mixed up. Again, remember that
we�re asking what function we differentiated to get the integrand.
We will be able to integrate the remaining four trig
functions in a couple of sections, but they all require the
Substitution Rule.
Now, let�s take care of exponential and logarithm
functions.
Integrating logarithms requires a topic that is usually
taught in Calculus II and so we won�t be integrating a logarithm in this class.
Also note the third integrand can be written in a couple of ways and don�t
forget the absolute value bars in the x in the answer to the third
integral.
Finally, let�s take care of the inverse trig and hyperbolic
functions.
As with logarithms integrating inverse trig functions
requires a topic usually taught in Calculus II and so we won�t be integrating
them in this class. There is also a different answer for the second integral
above. Recalling that since all we are asking here is what function did we
differentiate to get the integrand the second integral could also be,
Traditionally we use the first form of this integral.
Okay, now that we�ve got most of the basic integrals out of
the way let�s do some indefinite integrals. In all of these problems remember
that we can always check our answer by differentiating and making sure that we
get the integrand.
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