Remarks in italics are by me for extra explanation. These comments would not be necessary for full marks in an exam situation. Exercises taken from p.35 of https://jpmccarthymaths.wordpress.com/2011/02/01/math6037-general-information/
These questions all require integration by parts although they don’t explicitly say so.
Question 1
We must choose a and
— where we choose
according to the LIATE rule. Here there are no logs, no inverses but there is an algebraic (sum of powers of
). Hence let
,
(find
by integrating:
):
; and
.
Hence by the integration by parts formula:
We check our solution by differentiating (using the product rule):
,
as required.
Question 2
Again we must choose according to the LIATE rule. Hence we choose
(note that a strict application of the LIATE — choose the most complicated part — suggests
. Making this substitution (without applying the integration by parts formula) leads to the integral
— which needs the
substitution anyway…). Hence
(using
):
; and
.
Using the integration by parts formula:
.
We check our answer by differentiating :
,
as required.
Question 3
Rather than having to keep track of the limits we will first evaluate the indefinite integral:
By the LIATE rule, let and
:
; and
.
Hence by the integration by parts formula:
.
Now we need to evaluate this between the limits — and of course the constant of integration will cancel so we omit it:
.
Question 4
Again we compute the integral first without limits. Now by LIATE, let , so
:
and
Now using the integration by parts formula:
.
Now we need to evaluate this between the limits, again omitting the constant of integration:
.
Now and
for
,
:
,
with as required.
Question 5
Again by LIATE let ,
:
; and
.
Hence by the integration by parts formula:
.
Question 6
Again by LIATE we choose and
:
and
.
Hence by the integration by parts formula:
.
Now looking at the integral:
Can we do this with the integral tables, manipulation,substitution or by parts?? Whatever works will do — experience will tell us that we need to make a substition. By noticing the function-derivative pattern —
we should make the
‘function’ substitution
(LIATE will also tell us to pick this):
,
;
Question 7
Again by LIATE we choose and
:
and
.
Hence by the Integration by Parts formula:
Question 8
Again by LIATE let and
:
and
.
We should recognise (from the tables if need be) that is the derivative of
. Hence
. By the Integration by Parts formula:
.
where the integral of was taken from the tables.
Question 9
Apologies! Between these limits the area under the function is infinite — namely negative infinity as it is below the -axis. Look at this plot from Wolfram Alpha (an absolutely unbelievable piece of kit by the by):
http://www.wolframalpha.com/input/?i=Plot[Log[x]%2Fx^2%2C{x%2C0%2C2}]
Hence we’ll just evaluate without limits.
Now by the LIATE rule choose and
:
and
.
Hence by the Integration by Parts formula:
Question 10
Evaluating first without limits, by the LIATE rule let and
:
and
.
Hence by the Integration by Parts formula:
.
Now evaluating between the limits, dropping the constant of integration:
.
Now first off and
. Hence (or else using the calculator)
:
.
2 comments
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March 12, 2011 at 3:25 pm
Maris
I think that in question 8 the answer should be
theta*tan(theta)-log[sec(theta)]
instead of +log[sec(theta)]
[Corrected – J.P.]
March 12, 2011 at 3:51 pm
Maris
The last fraction in the answer in question 10 should be -28/9 instead of -23/9
[Corrected – J.P.]