INTEGRATION

NB: You have to do something about a product or a fraction.
Either they are in the tables or you let u = one part and du = (the rest)dx.

POWERS OF X

NB. , etc. Note that this rule does not apply to integrating .

Questions: 2005 I 8a(i), 2004 I 8a(i), 2003 I 8a (i), 2002 I 8a, 2001 I 8a (i), 2000 I 8a (i), 1999 I 8a, 1998 I 8a, 1996 I 8a (i), 1995 I 8a (i), 1994 I 8a.

SIMPLE SUBSTITUTIONS

(a) . Let u = ax and du = adx (i.e. .)

Questions: 2005 I 8a(ii), 2004 I 8a(ii), 2003 I 8a (ii), 2003 I 8c (i), 2001 I 8a (ii), 2000 I 8a (ii), 1997 I 8a (i), 1996 I 8a (ii), 1994 I 8a.

(b) . Let and

Questions: 1999 I 8b (ii),

EASY PRODUCTS

e.g. just square out the bracket
e.g.

Questions 1998 I 8b (ii), 1997 I 8a (ii), 1996 I 8a(ii)

EASY FRACTIONS

(a) e.g. just divide by the denominator to get

(b) e.g. . Let and

Harder Questions: 1997 I 8c, 1995 I 8c (ii), 1994 I 8c (i) first integral, 1994 I 8c (ii).

(c) e.g. . Let , , and .
Then square out the numerator and divide by the denominator.

Questions: 1997 I 8b (ii)

FRACTIONS WITH A QUADRATIC IN THE DENOMINATOR.

(a) or let

Note: n can be a two, three, or a half etc.

Questions: 2005 I 8b(i), 2003 I 8b (i), 2002 I 8b(i), 2001 I 8b (ii), 2000 I 8b (ii), 1998 I 8b (i),

Harder Questions: 1996 I 8c (i)

(b) or complete square for inverse tan /sin

Questions: 2004 I 8c(i), 2004 I 8b(i), 2002 I 8b (ii), 2001 I 8b (i), 2000 I 8c (i), 1995 I 8c (i), 1994 I 8c (i) first integral.

TRIGONOMETRIC INTEGRALS.

(a) Change a product to a sum using page 9 of the tables.

(b) and so on. This is a special case of (a). Use the formulae on page 9.

Questions: 2005 I 8b(ii), 2000 I 8b (i), 1999 I 8b (i), 1997 I 8b (i), 1996 I 8b (ii), 1995 I 8c (i)

(c) . Let
or . Let .

Questions: 2004 I 8b(ii), 2003 I 8b (ii),

AREA BETWEEN A CURVE AND A LINE

Questions: 2004 I 8c, 2002 I 8c, 2001 I 8c, 2000 I 8c (ii), 1998 I 8c, 1996 I 8c (ii), 1995 I 8b, 1994 I 8a.

AREA OF A CIRCLE

e.g.

Note: This is a rearrangement of , i.e. a circle of centre (0, 0) and radius 2.
The integral is the area of one quarter of the circle.
Use the double substitution and and change the resulting product to a sum using page 9 of the Tables.

Questions 1999 I 8c, 1996 I 8b (i),

VOLUMES OF ROTATION

For a curve rotated about the x-axis:

For a curve rotated about the y-axis:

Questions: 2005 I 8c(ii), 2003 I 8c (ii), 1995 I 8c (ii)