*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

Comments feed for this article

March 12, 2011 at 3:25 pm

MarisI 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

MarisThe last fraction in the answer in question 10 should be -28/9 instead of -23/9

[

Corrected – J.P.]