Calculate the following indefinite integrals! (hint: by parts)
Reminder
(2.1.2.1)
where the expression on the left is a function of x and the one on the right is a function of u , which can be converted after integration to a function of x .
The integration by parts generates from the rule of the differential of a product ... namely . By reshuffling the terms and integrating, one obtains:
(2.1.2.9)
The following example with uses d(lnx) in order to obtain a familiar integral:
(2.1.2.10)
The case of m=0 is included in (2.1.2.10) and was obtained by:
(2.1.2.11)
where u = lnx and v = x .
Tips. For a given integral the number of options for doing integration by parts is limited, therefore trial and error can be applied. On the other hand acquired experience is always helpful.
Parts 1-3
Solution of question 1
Similarly to (2.1.2.11), the integration by parts (2.1.2.9)
yields
The unsolved integral from part 1 needs a substitution
in order to obtain the final solution:
Parts 4-6
Solution of question 2
The solution of this question is very similar to the previous one. The parts
yield
By comparison with the solution of the previous question the substitution
gives the final solution
Parts 7-10
Solution of question 3
As already known there are integrals that can be solved by different approaches. This one is presented here by two alternative parts 7 and 8.
By following the steps of the previous questions one could use the same way of integration by parts:
which yields
Since the integral appearing on the left appears on the right with opposite sign, by reshufling the terms one obtains:
An alternative approach for parts 7 and 8 is
For the integration by parts one could use:
which yields
which is equivalent to the previous result of part 8.
In order to solve the integral from part 8 in terms of familiar integrals one can rewrite