Various Techniques

and

Hence or otherwise evaluate:

and

Evaluate also

STEP 2010, Math III, #8: Given that , write down an expression for

(i) By choosing the function to be of the form , find

Show that the choice of is not unique and, by comparing the two functions corresponding to two different values of ,

explain how the different choices are related.

(ii) Find the general solution of

and are related by , where and are constants and .

Show that

,

and hence integrate

Verify by direct integration that your result holds also in the case if but that your result does not hold in the case

if .

STEP 2006, Math III, #2: Let

and

where

.

(i) Show that

and hence that

.

(ii) Find

.

(iii) By considering , or otherwise, show that

(iv) Evaluate

in the case

.

STEP 2010, Math II, #2: Prove that

Find and prove a similar result for

in terms of .

(i) Let

Show that

for which he would obtain the correct value of .

STEP 2009, Math II, #5: Expand and simplify

(i) Evaluate

(ii) Find the total area between the curve

and the x-axis between the points and .

(iii) Evaluate

STEP 2009, Math II, #7: Let , where and are constants and is a non-zero constant.

Show that

where is a cubic polynomial.

Using this result, determine:

(i)

(ii)

(iii)

and . Starting with the expression

for the area swept out by OP, obtain the equivalent expression

(*)

The ends of a thin straight rod AB lie on a closed convex curve C. The point P on the rod is a fixed distance

from A and a fixed

distance

from B. The angle between AB and the positive direction is . As A and B move anticlockwise round C, the angle

increases from

to and P traces a closed convex curve D inside C, with the origin O lying inside D, as shown in the diagram.

Let be the coordinates of P. Write down the coordinates of A and B in terms of , , , and .

and , and that , show that

and find

Hence, or otherwise, integrate, for

,

(i)

(ii)

[You may use the results