*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 *https://jpmccarthymaths.wordpress.com/2011/02/01/math6037-general-information/

## Question 1

### (a)

*Now:*

and ,

*hence:*

### (b)

*There is no obvious anti-derivative and there is no obvious manipulation hence we are looking for the function-dervative pattern to make a substitution (or else use the LIATE rule). Notice that the top is the derivative of the bottom. Hence we will let be the *function; *i.e. the bottom:*

Let :

*Now put everything back into the integrand, suppressing the limits:*

*Now, *what do we have to raise to, to get ? *i.e. ; well . Hence :*

.

### (c)

*Again no obvious anti-derivative or manipulation hence we are looking at making a substitution.*

*There are a number of options here. One is the straight substitution . This will work out because , just a constant. So a function, and a multiple of it’s derivative are in the integrand. Alternatively the LIATE rule says pick . Finally there are a number of other options once we know about terms of the form . We will pursue this here.*

*Now , hence:
*

*Now is just a real number or scalar and because integration is essentially addition*, *we can take it out as a common term:*

*Now *

* (*)
*

*hence*

*Again ;*

## Question 2

### (a)

*According to the hint, we should choose :*

Let :

*Throwing this in, suppressing the limits for now:*

### (b)

*Now using (*):*

*using a calculator.*

## Question 3

**First off apologies: the middle term should have been .
**

*This is a tough question but if you think carefully about your original definition of integration as finding the area under a curve, then we have that if * *then:*

*When , (in yellow, blue and red respectively). Hence the if we find the area under these curves between any two points bigger than 1, the area under a ‘smaller’ function will be smaller than the area under a ‘bigger’ one.*

*This is what we have to do here. The problem is between what limits should be pick. If divine inspiration doesn’t help I suggest solving the general problem; i.e. just let and * *be anything (well careful: we want to integrate where , so we want and also ). Now writing the three functions in their power form (i.e. **):
*

* *

*To tidy up a little, using the log rule :*

*This is valid for all such that . How about ?*

## Question 4

### (a)

*Not in tables but try a manipulation:*

*If you are comfortable feel free to use the identity :*

:

.

### (b)

*No obvious anti-derivative or manipulation*. *Spot the function and derivative .*

Let :

*Again suppressing the limits*

*Now using the calculator, tables, or preferably, the unit circle, and :*

### (c)

*There is no obvious anti-derivative (direct integration by rule) so we’re looking at a manipulation or a substitution. The trick here is to know that we know how to integrate:*

,

*and because ** is of the form* , *we can *complete the square *and write * *and make the substitution *.

Now find such that

Compare coefficients:

We need

and

Hence and . Now:

Let :

*Again suppressing the limits:*

*Now using a calculator, tables or preferably a little 30-60-90 right-angle-triangle diagram:*

## Question 5

*Differentiation is *linear. *Let :*

*Hence*

*There are two ways to finish this question. One is to note that, by the Chain Rule:*

*so * *has anti-derivative . This is the obvious way.*

*Alternatively rewrite the integral:*

*.*

*Now looking for the function-derivative pattern, let *:

*Suppressing the limits:*

*The ‘2’s cancel. Putting in the limits:*

*Now some facts about exponentials and logs:*

*,*

*,*

*.*

*Hence, with (multiply above and below by 2):*

*.*

## Question 6

### (a)

*We want to find such that*

* and *

* and *hence* . *

*So . Now considering the integral*

*,*

*make the substitution :*

*:*

*Now using the tables:*

*(b)*

*Apologies – this question requires partial fractions – which we didn’t do until Week 2. *

**Question 7**

*Here we used the fact that . This is a general rule for real numbers: .*

*Now, using this expansion and the fact that integration is linear (i.e. we can integrate term by term):*

*What function when differentiated gives ? The answer is : .*

*This is not in the tables. There is no obvious manipulation. Hence we may need a substitution. So looking for the function-derivative pattern, we see that has derivative so we let “function”; (LIATE will also give you this – LIATE says takes the first thing on the list – the more complicated the better – and is more complicated that ):*

*.*

*Putting this back into :*

*Again there is no obvious rule or manipulation so we are looking for a substitution. Convince yourself that is the appropriate substitution…*

*Hence*

*Now throwing this back into the original integral:*

*as is just some constant, which we can rename for short.*

## Question 8

*Not in the tables and no obvious manipulation. Again, convince yourself (function-derivative pattern or LIATE) that is an appropriate substitution:*

*.*

*Putting back into the integral and suppressing the limits:*

*.*

*Now one of the facts about logarithms is that:*

*;*

*.
*

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